Lecture Notes on

C Algebras and Quantum Mechanics

Draft April

NP Landsman

Kortewegde Vries Institute for Mathematics University of Amsterdam

Plantage Muidergracht

TV AMSTERDAM THE NETHERLANDS

email nplwinsuvanl

homepage httpturingwinsuvanl npl

telephone

oce Euclides a

CONTENTS

Contents

Historical notes

Origins in functional analysis and quantum mechanics

Rings of op erators von Neumann algebras

Reduction of unitary group representations

The classication of factors

C algebras

Elementary theory of C algebras

Basic denitions

Banach algebra basics

CommutativeBanach algebras

Commutative C algebras

Sp ectrum and functional calculus

Positivityin C algebras

Ideals in C algebras

States

Representations and the GNSconstruction

The GelfandNeumark theorem

Complete p ositivity

Pure states and irreducible representations

The C algebra of compact op erators

The double commutant theorem

Hilb ert C mo dules and induced representations

Vector bundles

Hilb ert C mo dules

The C algebra of a Hilb ert C mo dule

Morita equivalence

Rieel induction

The imprimitivity theorem

Group C algebras

C dynamical systems and crossed pro ducts

Transformation group C algebras

The abstract transitive imprimitivit ytheorem

Induced group representations

Mackeys transitive imprimitivity theorem

Applications to quantum mechanics

The mathematical structure of classical and quantum mechanics

Quantization

Stinesprings theorem and coherent states

Covariant lo calization in conguration space

Covariantquantization on phase space

Literature

CONTENTS

Historical notes

Origins in functional analysis and quantum mechanics

The emergence of the theory of op erator algebras may b e traced back to at least three develop

ments

The work of Hilb ert and his pupils in Gottingen on integral equations sp ectral theory and

innitedimensional quadratic forms

The discovery of quantum mechanics by Heisenb erg in Gottingen and indep endently

bySchrodinger in Zuric h

The arrival of John von Neumann in Gottingen to b ecome Hilb erts assistant

Hilb erts memoirs on integral equations app eared b etween and In his studentE

Schmidt dened the space in the mo dern sense F Riesz studied the space of all continuous linear

maps on and various examples of L spaces emerged around the same time However

the abstract concept of a Hilb ert space was still missing

Heisenb erg discovered a form of quantum mechanics which at the time was called matrix

hrodinger was led to a dierent formulation of the theory which he called wave mechanics Sc

mechanics The relationship and p ossible equivalence b etween these alternativeformulations of

quantum mechanics which at rst sight lo oked completely dierent was much discussed at the

time It was clear from either approach that the b o dy of work mentioned in the previous paragraph

was relevanttoquantum mechanics

Heisenb ergs pap er initiating matrix mechanics was followed by the Dreimannerarb eit of Born

Heisenb erg and Jordan all three were in Gottingen at that time Born was one of the

few physicists of his time to be familiar with the concept of a matrix in previous research he

had even used innite matrices Heisenbergs fundamental equations could only be satised by

innitedimensional matrices Born turned to his former teacher Hilb ert for mathematical advice

Hilb ert had been interested in the mathematical structure of physical theories for a long time

his Sixth Problem called for the mathematical axiomatization of physics Aided by his

assistants Nordheim and von Neumann Hilb ert thus ran a seminar on the mathematical structure

of quantum mechanics and the three wrote a joint pap er on the sub ject now obsolete

It was von Neumann alone who at the age of saw his way through all structures and

mathematical diculties In a series of pap ers written b etween culminating in his b o ok

Mathematische Grund lagen der Quantenmechanik he formulated the abstract concept of a

Hilb ert space develop ed the sp ectral theory of b ounded as wellasunb ounded normal op erators

on a Hilb ert space and proved the mathematical equivalence b etween matrix mechanics and wave

mechanics Initiating and largely completing the theory of selfadjoint op erators on a Hilb ert

space and intro ducing notions such as density matrices and quantum entropy this b o ok remains

the denitive account of the mathematical structure of elementary quantum mechanics von

Neumanns book was preceded by Diracs The Principles of Quantum Mechanics which

contains a heuristic and mathematically unsatisfactory accountofquantum mechanics in terms of

linear spaces and op erators

Rings of op erators von Neumann algebras

In one of his pap ers on Hilb ert space theory von Neumann denes a ring of op erators

M nowadays called a von Neumann algebra as a subalgebra of the algebra BH of all

b ounded op erators on a Hilb ert space H ie a subalgebra which is closed under the involution

A A that is closed ie sequentially complete in the weak op erator top ology The latter may

b e dened by its notion of convergence a sequence fA g of b ounded op erators weakly converges

n

to A when A A for all H This typeofconvergence is partly motivated by

n

quantum mechanics in whichA is the exp ectation value of the observable A in the state

t and has unit norm provided that A is selfadjoin

HISTORICAL NOTES

For example BH is itself a von Neumann algebra Since the weak top ology is weaker than

the uniform or norm top ology on BH avon Neumann algebra is automatically normclosed

as well so that in terminology to b e intro duced later on a von Neumann algebra b ecomes a C

algebra when one changes the top ology from the weak to the uniform one However the natural

top ology on a von Neumann algebra is neither the weak nor the uniform one

In the same pap er von Neumann proves what is still the basic theorem of the sub ject a

subalgebra M of BH containing the unit op erator Iis weakly closed i M M Here the

commutant M of a collection M of b ounded op erators consists of all b ounded op erators which

commute with all elements of M and the bicommutant M is simply M This theorem is

remarkable in relating a top ological condition to an algebraic one one is reminded of the much

simpler fact that a linear subspace K of H is closed i K where K is the orthogonal complement

of K in H

Von Neumanns motivation in studying rings of op erators was plurifold His primary motivation

probably came from quantum mechanics unlike many physicists then and even now he knew

that all Hilb ert spaces of a given dimension are isomorphic so that one cannot characterize a

physical system bysaying that its Hilb ert space of pure states is L R Instead von Neumann

quantummechanical systems by algebraic conditions on the observables hop ed to characterize

This programme has to some extent b een realized in algebraic quantum eld theory Haag and

followers

Among von Neumanns interest in quantum mechanics was the notion of entropy he wished

n

to dene states of minimal information When H C for n such a state is given by the

density matrix Inbut for innitedimensional Hilb ert spaces this state may no longer be

dened Density matrices maybe regarded as states on the von Neumann algebraBH in the

sense of p ositive linear functionals which map I to As we shall see there are von Neumann

algebras on innitedimensional Hilb ert spaces which do admit states of minimal information that

generalize In viz the factors of typ e I I see b elow



Furthermore von Neumann hop ed that the divergences in quantum eld theory mightbere

moved by considering algebras of observables dierentfromBH This hop e has not materialized

although in algebraic quantum eld theory the basic algebras of lo cal observables are indeed not

of the form BH but are all isomorphic to the unique hyp ernite factor of typ e I I I see b elow



Motivation from a dierent direction came from the structure theory of algebras In the present

context a theorem of Wedderburn says that a von Neumann algebra on a nitedimensional Hilb ert

space is isomorphic to a direct sum of matrix algebras Von Neumann wondered if this or a

similar result in which direct sums are replaced by direct integrals see below still holds when

the dimension of H is innite As we shall see it do es not

ation came from group representations Von Neumanns bicom Finallyvon Neumanns motiv

mutant theorem implies a useful alternativecharacterization of von Neumann algebras from now

on we add to the denition of a von Neumann algebra the condition that M contains I

The commutant of a group U of unitary op erators on a Hilb ert space is a von Neumann algebra

and converselyevery von Neumann algebra arises in this way In one direction one trivially veries

that the commutantofany set of b ounded op erators is weakly closed whereas the commutantof

a set of b ounded op erators which is closed under the involution is a algebra In the opp osite

direction given M one takes U to b e the set of all unitaries in M

This alternativecharacterization indicates whyvon Neumann algebras are imp ortantinphysics

the set of b ounded op erators on H which are invariant under a given group representation U G

on H is automatically a von Neumann algebra Note that a given group U of unitaries on H may

b e regarded as a representation U of U itself where U is the identitymap

Reduction of unitary group representations

The p ossible reduction of U G is determined by the von Neumann algebras U G and U G

For example U is irreducible i U G CI Schurs lemma The representation U is called

primary when U G has a trivial center that is when U G U G CI When G is

compact so that U is discretely reducible this implies that U is a multiple of a xed irreducible

Reduction of unitary group representations

representation U on a Hilb ert space H sothat HH KandU U I

K

When G is not compact but still assumed to b e lo cally compact unitary representations may

be reducible without containing any irreducible subrepresentation This o ccurs already in the

simplest p ossible cases such as the regular representation of G R on H L R that is one

puts U xy y x The irreducible wouldb e subspaces of H would be spanned by the

vectors y expipy but these functions do not lie in L R The solution to this problem

p

yvon Neumann in a pap er published in but written in the thirties the ideas in was given b

it must have guided von Neumann from at least on

Instead of decomp osing H as a direct sum one should decomp ose it as a direct integral

To do so one needs to assume that H is separable This means that rstly one has a measure

space and a family of Hilb ert spaces fH g A section of this family is a function



fH g for which H To dene the direct integral of the H with resp ect to



the measure one needs a sequence of sections f g satisfying the two conditions that rstly the

n

function b e measurable for all n m and secondly that for eachxed the

n m

span H There then exists a unique maximal linear subspace of the space of all sections

n 

whichcontains all and for which all sections are measurable

n

For it then makes sense to dene



Z

d



The direct integral

Z

d H



is then by denition the subset of of functions for which When is discrete



the direct integral reduces to a direct sum

An op erator A on this direct integral Hilb ert space is said to b e diagonal when

AA

for some suitably measurable family of op erators A on H We then write

Z

A d A



Thus a unitary group representation U GonH is diagonal when

U xU x

for all x G in which case we of course write

Z

d U U



Reducing a given representation U on some Hilb ert space then amounts to nding a unitary map

V between H and some direct integral Hilb ert space such that each H carries a representation

U andVUxV is diagonal in the ab ove sense with A U x When H is separable one may

ys reduce a unitary representation in suchaway that the U o ccurring in the decomp osition alwa

are primary and this central decomp osition of U is essentially unique

To completely reduce U one needs the U to be irreducible so that is the space G of all

equivalence classes of irreducible unitary representations of G Complete reduction therefore calls

for a further direct integral decomp osition of primary representations this will b e discussed b elow

For example one maytakeR with Leb esgue measure and take the sequence f g to

n

consist of a single strictly p ositive measurable function This leads to the direct integral decom

p osition

Z

L R dp H

p R

HISTORICAL NOTES

in whicheach H is C To reduce the regular representation of R on L R one simply p erforms

p

aFourier transform V L R L R ie

Z

ipy

V p dy e y

R

This leads to VUxV pexpipxp so that U has b een diagonalized the U xabove are

now the onedimensional op erators U x expipx on H C Wehave therefore completely

p p

reduced U

As far as the reduction of unitary representations is concerned there exist two radically dierent

classes of lo cally compact groups the class of all lo cally compact groups includes for example

all nitedimensional Lie groups and all discrete groups A primary representation is said to b e

of typ e I when it may b e decomp osed as the direct sum of irreducible subrepresentations these

subrepresentations are necessarily equivalent A lo cally compact group is said to be typ e I or

tame when every primary representation is a multiple of a xed irreducible representation in

other words a group is typ e I when all its primary representations are of typeIIfnotthegroup

is called nontyp e I or wild An example of a wild group well known to von Neumann is the free

group on two generators Another example discovered at a later stage is the group of matrices of

the form

it

e z

it

A

e w

is an irrational real number t R and z w C where

When G is wild curious phenomena may o ccur By denition a wild group has primary unitary

representations which contain no irreducible subrepresentations More bizarrely representations

of the latter typ e may b e decomp osed in two alternativeways

Z Z

d U d U U



 

G G

where the measures and are disjoint that is supp orted by disjoint subsets of G



A reducible primary representation U mayalways b e decomp osed as U U U In case

h h

that U is not equivalentto U andU is not of typ e I it is said to b e a representation of typ e I I

h

When U is neither of typ e I nor of typ e I I it is of typ e IIIInthatcaseU is equivalenttoU

h

indeed all prop er subrepresentations of a primary typ e I I I representation are equivalent

The classication of factors

Between and von Neumann wrote lengthy dicult and profound pap ers of which

were in collab oration with Murray in which the study of his rings of op erators was initiated

According to IE Segal these pap ers form p erhaps the most original ma jor work in mathematics

in this century

The analysis of Murray and von Neumann is based on the study of the pro jections in a von

Neumann algebra M a pro jection is an op erator p for which p p p indeed M is generated

They noticed that one may dene an equivalence relation on the set of all by its pro jections

pro jections in M in which p q i there exists a partial isometry V in M suchthat V V p

and VV q When M BH the op erator V is unitary from pH to q H and annihilates pH

Hence when M BH one has p q i pH and q H have the same dimension for in that case

one maytakeany V with the ab ove prop erties

An equivalentcharacterization of arises when we write M U G for some unitary represen

tation U ofagroupG as wehave seen this always applies then p q i the subrepresentations

pU and qU on pH and q H resp ectively are unitarily equivalent

Moreover Murrayand von Neumann dene a partial orderering on the collection of all pro

jections in M by declaring that p q when pq p that is when pH q H This induces a

partial orderering on the set of equivalence classes of pro jections by putting p q when the

The classication of factors

equivalence classes p and q contain representatives p and q such that p q For M BH

this actually denes a total ordering on the equivalence classes in which p q when pH has

the same dimension as q H aswe just saw this is indep endent of the choice of p pand q q

More generally Murrayandvon Neumann showed that the set of equivalence classes of pro jec

tions in M is totally ordered by whenever M is a factor Avon Neumann algebra M is a factor

when M M CIwhen M U G this means that M is a factor i the representation U is

primary The study of von Neumann algebras acting on separable Hilb ert spaces H reduces to the

study of factors for von Neumann proved that every von Neumann algebra M BH maybe

uniquely decomp osed as in

Z

d H H



Z

d M M



where almost each M is a factor For M U G the decomp osition of H amounts to the

central decomp osition of U G

As we ha ve seen for the factor M BH the dimension d of a pro jection is a complete

invariant distinguishing the equivalence classes p The dimension is a function from the set of



all pro jections in BH toR satisfying

dp whenp and d

dpdq ip q

dp q dpdq when pq ie when pH and q H are orthogonal

dp i p is nite

Here a pro jection in BH is called nite when pH is nitedimensional Murray and von Neumann

nowproved that on any factor M acting on a separable Hilb ert space there exists a function d



from the set of all pro jections in M to R satisfying the ab ove prop erties Moreover d

is unique up to nite rescaling For this to be the p ossible Murrayandvon Neumann dene a

pro jection to b e nite when it is not equivalenttoany of its prop er subpro jections an innite

pro jection is then a pro jection which has prop er subpro jections to which it is equivalent For

M BH this generalized notion of niteness coincides with the usual one but in other factors

all pro jections may b e innite in the usual sense yet some are nite in the sense of Murray and

von Neumann One may say that in order to distinguish innitedimensional but inequivalent

pro jections the dimension function d is a renormalized version of the usual one

A rst classication of factors on a separable Hilb ert space is now p erformed by considering

the p ossible niteness of its pro jections and the range of d A pro jection p is called minimal or

atomic when there exists no qpie q p and q p One then has the following p ossibilities

for a factor M

typ e I where n M has minimal pro jections all pro jections are nite and d takes

n

the values f ng A factor of typ e I is isomorphic to the algebra of n n matrices

n

typ e I M has minimal pro jections and d takes the values f g Such a factor is

H for separable innitedimensional H isomorphic to B

typ e I I M has no minimal pro jections all pro jections are innitedimensional in the usual



sense and I is nite Normalizing d suchthat dI the range of d is the interval

typ e II M has no minimal pro jections all nonzero pro jections are innitedimensional

in the usual sense but M has nitedimensional pro jections in the sense of Murray and von

Neumann and I is innite The range of d is

HISTORICAL NOTES

typ e III M has no minimal pro jections all nonzero pro jections are innitedimensional

and equivalent in the usual sense as well as in the sense of Murrayandvon Neumann and d

assumes the values f g

With M U G where as wehave seen the representation U is primary i M is a factor U

is of a given typ e i M is of the same typ e

One sometimes says that a factor is nite when I is nite so that dI hence typ e I

n

and typ e I I factors are nite Factors of typ e I and I I are then called seminite and typ e



I I I factors are purely innite

It is hard to construct an example of aII factor and even harder to write do wn a type III



factor Murrayand von Neumann managed to do the former and von Neumann did the latter

by himself but only years after he and Murray had recognized that the existence of typ e III

factors was a logical p ossibility However they were unable to provide a further classication of

all factors and they admitted having no to ols to study typ e I I I factors

Von Neumann was fascinated by II factors In view of the range of d he b elieved these



dened some form of continuous geometry Moreover the existence of a I I factor solved one of



the problems that worried him in quantum mechanics For he showed that on a II factor M



the dimension function d dened on the pro jections in Mmaybe extended to a p ositive linear

functional tr on M with the prop ertythat tr UAU tr A for all A M and all unitaries U in

M This trace satises tr I dI and gavevon Neumann the state of minimal information

he had sought Partly for this reason he b elieved that physics should b e describ ed byII factors



At the time not manypeoplewere familiar with the dicult pap ers of Murrayandvon Neu

mann and until the sixties only a handful of mathematicians worked on op erator algebras eg

Segal Kaplansky Kadison Dixmier Sakai and others The precise connection b etween von Neu

mann algebras and the decomp osition of unitary group representations envisaged byvon Neumann

was worked out byMackey Mautner Go dement and AdelsonVelskii

In the sixties a group of physicists led by Haag realized that op erator algebras could b e a useful

to ol in quantum eld theory and in the quantum statistical mechanics of innite systems This has

led to an extremely fruitful intercation b etween physics and mathematics whichhashelpedboth

sub jects In particular in Haag observed a formal similarity between the collection of all

von Neumann algebras on a Hilb ert space and the set of all causally closed subsets of Minkowksi

spacetime Here a region O in spacetime is said to b e causally closed when O O where

O consists of all p oints that are spacelike separated from O The op eration OO on causally

closed regions in spacetime is somewhat analogous to the op eration M M on von Neumann

algebras Thus Haag prop osed that a quantum eld theory should b e dened byanet of lo cal

observablesthisisamapO MO from the set of all causally closed regions in spacetime

to the set of all von Neumann algebras on some Hilb ert space such that MO MO when



O O andM O MO



This idea initiated algebraic quantum eld theory a sub ject that really got o the ground

with pap ers by Haags pupil Araki in and by Haag and Kastler in From then till

the present day algebraic quantum eld theory has attracted a small but dedicated group of

mathematical physicists One of the result has b een that in realistic quantum eld theories the

lo cal algebras MO must all b e isomorphic to the unique hyp ernite factor of typ e I I I discussed



below Hence von Neumanns b elief that physics should use I I factors has not b een vindicated



A few years later an extraordinary coincidence to ok place which was to play an es

sential role in the classication of factors of typ e I I I On the mathematics side Tomita develop ed

atechnique in the study of von Neumann algebras which nowadays is called mo dular theory

or TomitaTakesaki theory apart from clarifying Tomitas work Takesaki made essential con

tributions to this theory Among other things this theory leads to a natural timeevolution on

certain factors On the physics side Haag Hugenholtz and Winnink characterized states of ther

mal equilibrium of innite quantum systems by an algebraic condition that had previously b een

intro duced in a heuristic setting by Kub o Martin and Schwinger and is therefore called the KMS

condition This condition leads to typ e I I I factors equipp ed with a timeevolution which coincided

with the one of the TomitaTakesaki theory

In the hands of Connes the TomitaTakesaki theory and the examples of typ e III factors

C algebras

provided byphysicists Araki Woods Powers and others eventually led to the classication of all

hyp ernite factors of typ e I I and I I I the complete classication of all factors of typ e I is already

given by the list presented earlier These are factors containing a sequence of nitedimensional

subalgebras M M M such that M is the weak closure of M Exp erience shows

 n n

that all factors playing a role in physics are hyp ernite and many natural examples of factors

constructed by purely mathematical techniques are hyp ernite as well The work of Connes for

whichhewas awarded the Fields Medal in and others led to the following classication of

hyp ernite factors of typ e I I and I I I up to isomorphism

There is a unique hyp ernite factor of typ e II In physics this factor o ccurs when one



considers KMSstates at innite temp erature

There is a unique hyp ernite factor of typ e I I namely the tensor pro duct of the hyp ernite

II factor with BK for an innitedimensional separable Hilb ert space K

There is a family of typ e I I I factors lab eled by For the factor of type III

is unique There is a family of type III factors which in turn is has b een classied in terms



of concepts from ergo dic theory

As wehave mentioned already the unique hyp ernite I I I factor plays a central role in algebraic



quantum eld theory The unique hyp ernite I I factor was crucial in a sp ectacular development



in which the theory of inclusions of I I factors was related to knot theory and even led to a new



knot invariant In Jones was awarded a Fields medal for this work the second one to b e given

to the once obscure eld of op erator algebras

C algebras

In the midst of the Murrayvon Neumann series of pap ers Gelfand initiated a separate develop

ment combining op erator algebras with the theory of Banac h spaces In he dened the con

cept of a Banach algebra in whichmultiplication is separately continuous in the normtop ology

He pro ceeded to dene an intrinsic sp ectral theory and proved most basic results in the theory of

commutative Banach algebras

In Gelfand and Neumark dened what is now called a C algebra some of their axioms

were later shown to b e sup eruous and proved the basic theorem that each C algebra is isomor

phic to the normclosed algebra of op erators on a Hilb ert space Their pap er also contained the

rudiments of what is now called the GNS construction connecting states to representations In

its present form this construction is due to Segal a great admirer of von Neumann who

generalized von Neumanns idea of a state as a p ositive normalized linear functional from BH to

arbitrary C algebras Moreover Segal returned to von Neumanns motivation of relating op erator

algebras to quantum mechanics

As with von Neumann algebras the sixties brought a fruitful interaction b etween C algebras

and quantum physics Moreover the theory of C algebras turned out to b e interesting b oth for

intrinsic reasons structure and representation theory of C algebras as well as b ecause of its

connections with a number of other elds of mathematics Here the strategy is to take a given

mathematical structure try and nd a C algebra which enco des this structure in some way and

then obtain information ab out the structure through proving theorems ab out the C algebra of

the structure

The rst instance where this led to a deep result which has not b een proved in any other

way is the theorem of Gelfand and Raikov stating that the unitary representations of a

lo cally compact group separate the p oints of the group that is for each pair x y there exists a

y This was proved by constructing a C algebra unitary representation U for which U x U

C G of the group G showing that representations of C G bijectively corresp ond to unitary

representations of G and nally showing that the states of an arbitrary C algebra A separate the

elements of A

Other examples of mathematical structures that may b e analyzed through an appropriate C

algebra are group actions group oids foliations and complex domains The same idea lies at the

ELEMENTARY THEORYOF C ALGEBRAS

basis of noncommutative geometry and noncommutative top ology Here the starting

point is another theorem of Gelfand stating that any commutative C algebra with unit is

isomorphic to C X where X is a compact Hausdor space The strategy is now that the basic to ols

in the top ology of X and when appropriate in its dierential geometry should b e translated into

to ols p ertinent to the C algebra C X and that subsequently these to ols should b e generalized

to noncommutative C algebras

This strategy has b een successful in K theory whose noncommutativeversion is even simpler

than its usual incarnation and in de Rham cohomology theory whose noncommutativeversion

is called cyclic cohomology Finally homology cohomology K theory and index theory haven

b een unied and made noncommutativeinthe KKtheory of Kasparov The basic to ol in KK

theory is the concept of a Hilb ert C mo dulewhichwe will study in detail in these lectures

Elementary theory of C algebras

Basic denitions

All vector spaces will be dened over C and all functions will be C valued unless we explicitly

state otherwise The abbreviation i means if and only if which is the same as the symbol

An equation of the typ e a b means that a is by denition equal to b

Denition A norm onavector space V is a map kk VR such that

k v k for al l v V

k v k i v

k v k jj k v k for al l C and v V

k v w kk v k k w k triangle inequality

Anormon V denes a metric d on V by dv wk v w k Avector space with a norm which

is complete in the associated metric in the sense that every Cauchy sequenceconverges is cal led

a Banach space We wil l denote a generic Banach space by the symbol B

The two main examples of Banach spaces we will encounter are Hilb ert spaces and certain

collections of op erators on Hilb ert spaces

isamap VV C such that Denition A preinner pro duct onavector space V

v v w w v w v w v w v w for

           

al l C and v v w w V

   

v v for al l v V

Anequivalent set of conditions is

v ww v for al l v w V

v w w v w v w for al l C and v w w V

     

v v for al l v V

Apreinner product for which v v i v is cal ledaninner pro duct

The equivalence between the two denitions of a preinner pro duct is elementary in fact to

derive the rst axiom of the second characterization from the rst set of conditions it is enough

v iw Either way one derives the to assume that v v R for all v use this realitywith v

CauchySchwarz inequality

jv wj v vw w

Basic denitions

for all v w V Note that this inequality is valid even when is not an inner pro duct but

merely a preinner pro duct

It follows from these prop erties that an inner pro duct on V denes anormonV by k v k

p

v v the triangle inequality is automatic

Denition A Hilb ert space is a vector space with inner product which is complete in the

associated norm We wil l usual ly denote Hilbert spaces by the symbol H

A Hilb ert space is completely characterized by its dimension ie by the cardinality of an

arbitrary orthogonal basis To obtain an interesting theory one therefore studies op erators on a

Hilb ert space rather than the Hilb ert space itself To obtain a satisfactory mathematical theoryit

is wise to restrict oneself to b ounded op erators We recall this concept in the more general context

of arbitrary Banach spaces

Denition A b ounded op erator on a Banach space B is a linear map A B B for

which

k A k sup fk Av k j v B k v kg

The number k A k is the op erator norm or simply the norm of A This terminology is

justied as it follows almost immediately from its denition and from the prop erties of the the

norm on B that the op erator norm is indeed a norm

It is easily shown that a linear map on a Banach space is continuous i it is a b ounded op erator

but we will never use this result Indeed in arguments involving continuous op erators on a Banach

space one almost always uses b oundedness rather than continuity

When B is a Hilb ert space H the expression b ecomes

k A k sup fAA j H g

When A is b ounded it follows that

k Av kk A kkv k

for all v B Conversely when for A there is a C suchthat k Av k C k v k for all v

then A is b ounded with op erator norm k A k equal to the smallest p ossible C for whichtheabove

inequality holds

Prop osition The space BB of al l bounded operators on a Banach space B is itself a

Banach spaceintheoperator norm

In view of the comments following it only remains to b e shown that BB is complete in

the op erator norm Let fA g be a Cauchy sequence in BB In other words for any there

n

is a natural number N such that k A A k when n m N For arbitrary v B the

n m

sequence fA v g is a Cauchy sequence in B b ecause

n

k A v A v kk A A kk v k k v k

n m n m

for n m N Since B is complete by assumption the sequence fA v g converges to some w B

n

B by Av w lim A v This map is obviously linear Taking n Now dene a map A on

n n

in we obtain

k Av A v k k v k

m

for all m N and all v B It now follows from that A A is b ounded Since

m

A A A A and BB is a linear space we infer that A is b ounded Moreover and

m m

erges to A Since wehave imply that k A A k for all mN so that fA g conv

m n

just seen that A BB this proves that BB is complete

We dene a functional on a BanachspaceB as a linear map BC which is continuous

in that v j C k v k for some C andallv B The smallest such C is the norm

k k sup fjv jvB k v kg

ELEMENTARY THEORYOF C ALGEBRAS

The dual B of B is the space of all functionals on B Similarly to the pro of of one shows that

B is a Banach space For later use we quote without pro of the fundamental HahnBanach

theorem

Theorem For a functional on a linear subspace B of a Banach space B there exists a

 

functional on B such that on B and k kk k In other words each functional dened

  

on a linear subspaceof B has an extension to B with the same norm

Corollary When v for al l B then v

For v wemay dene a functional on C v by v and extend it to a functional

 

on B with norm

Recall that an algebra is a vector space with an asso ciative bilinear op eration multiplication

A A Awe usually write AB for A B It is clear that BB is an algebra under op erator

multiplication Moreover using twice for each v B one has

k AB v kk A kk Bv kk A kk B kk v k

Hence from we obtain k AB kk A kk B k

A which is at the same time an algebra Denition A Banachalgebrais a Banach space

in which for al l A B A one has

k AB kk A kk B k

It follows that multiplication in a Banach algebra is separately continuous in eachvariable

As wehave just seen for any Banach space B the space BB of all b ounded op erators on B is

a Banach algebra In what follows we will restrict ourselves to the case that B is a Hilb ert space

H this leads to the Banach algebra BH This algebra has additional structure

Denition An involution on an algebra A is a reallinear map A A such that for al l

A B A and C one has

A A

AB B A

A A

A algebra is an algebra with an involution

The op erator adjoint A A on a Hilb ert space dened by the prop ertyA A

denes an involution on BH Hence BH isa algebra As in this case an element A of a C

algebra A is called selfadjoint when A Awe sometimes denote the collection of all selfadjoint

elements by

A fA Aj A Ag

R

Since one may write

A A A A

A A iA i

i

every elementofA is a linear combination of two selfadjointelements

To see how the norm in BH is related to the involution we pickH and use the Cauchy

Schwarz inequality and to estimate

k A k AAA A k kk A A kk A A kk k

Using and we infer that

k A k k A A kk A kk A k

kk A k so that This leads to k A kk A k Replacing A by A and using yields k A

k A kk A k Substituting this in we derive the crucial prop erty k A A kk A k

This motivates the following denition

Banach algebra basics

Denition A C algebra is a complex Banach space A which is at the same time a

algebra such that for al l A B A one has

k AB k k A k k B k

k A A k k A k

In other words a C algebra isaBanach algebra in which holds

Here a Banach algebra is of course a Banach algebra with involution Combining and

one derives k A kk A k as in the preceding paragraph we infer that for all elements A

of a C algebra one has the equality

k A kk A k

The same argument proves the following

Lemma A Banach algebra in which k A k k A A k is a C algebra

We have just shown that BH is a C algebra Moreover each op erator normclosed

algebra in BH is a C algebra by the same argument A much deep er result which we will

formulate precisely and prove in due course states the converse of this each C algebra is isomor

phic to a normclosed algebra in BH for some Hilb ert space H Hence the axioms in

characterize normclosed algebras on Hilb ert spaces although the axioms make no reference to

Hilb ert spaces at all

For later use we state some selfevident denitions

Denition A morphism between C algebras A B is a complex linear map A B

such that

AB AB

A A

for al l A B A An isomorphism is a bijective morphism Two C algebras are isomorphic

when there exists an isomorphism between them

One immediately checks that the inverse of a bijective morphism is a morphism It is remark

able however that an injectiv e morphism and hence an isomorphism between C algebras is

automatically isometric For this reason the condition that an isomorphism be isometric is not

included in the denition

Banach algebra basics

The material in this section is not included for its own interest but b ecause of its role in the

theory of C algebras Even in that sp ecial context it is enlightening to see concepts suchasthe

sp ectrum in their general and appropriate setting

Recall Denition A unit in a Banach algebra A is an element I satisfying IA AI A

for all A A and

k I k

I where z C Note that in A Banach algebra with unit is called unital We often write z for z

a C algebra the prop erty IA AI A already implies take A I so that I I I

taking the adjoint this implies I I so that follows from

When a Banach algebra A do es not contain a unit we can always add one as follows Form

the vector space

A A C

I

and makethisinto an algebra by means of

A IB I AB B A I

ELEMENTARY THEORYOF C ALGEBRAS

where wehave written A I for A etc In other words the number in C is identied with

IFurthermore dene a norm on A by

I

k A I kk A k jj

In particular k I k Using in Aaswell as one sees from and that

I kk B I k k A IB I kk A kk B k jj k B k jj k A k jjjj k A

so that A is a Banach algebra with unit Since by the norm of A A in A coincides with

I

the norm of A I in A wehaveshown the following

I

Prop osition For every Banach algebra without unit there exists a unital Banach algebra A

I

and an isometric hence injective morphism A A such that A A C

I I

As we shall see at the end of section the unitization A with the given prop erties is not

I

unique

Denition Let A bea unital Banach algebra The resolvent A of A A is the set of

al l z C for which A z I has a twosided inverse in A

The sp ectrum A of A A is the complement of A in C in other words A is the set

of al l z C for which A z I has no twosided inverse in A

When A has no unit the resolvent and the spectrum aredenedthrough the embedding of A in

A A C

I

When A is the algebra of n n matrices the sp ectrum of A is just the set of eigenvalues

For A BH Denition repro duces the usual notion of the sp ectrum of an op erator on a

Hilb ert space

When A has no unit the sp ectrum AofA A always contains zero since it follows from

that A never has an inverse in A

I

Theorem The spectrum A of any element A of a Banach algebrais

contained in the set fz C jjz jkA kg

compact

not empty

The pro of uses two lemmas We assume that A is unital

P

n

k 

Lemma When k A k the sum A converges to I A

k 



Hence A z I always exists when jz j k A k

We rst show that the sum is a Cauc hy sequence Indeed for nm one has

n m n n n

X X X X X

k k k k k

k A A kk A k k A k k A k

k  k  k m k m k m

For n m this go es to by the theory of the geometric series Since A is complete the Cauchy

P

n

k

sequence A converges for n Now compute

k 

n n

X X

n k k  k

I A A A A I A

k  k 

Hence

n

X

k n n

k I A I A kk A kk A k

k 

Banach algebra basics

which forn ask A k by assumption Thus

n

X

k

lim A I AI

n

k 

By a similar argument

n

X

k

lim I A A I

n

k 

so that by continuityofmultiplication in a Banach algebra one nally has

n

X

k 

lim A I A

n

k 

  

The second claim of the lemma follows b ecause A z z I Az which exists

b ecause k Az k when jz j k A k

To prove that A is compact it remains to b e shown that it is closed

Lemma The set



GAfA Aj A existsg

of invertible elements in A is open in A

 

Given A GA takea B A for which k B k k A k By this implies

 

k A B kk A kkB k

   

I A B hasaninverse namely I A B A whichexistsby and Hence A B A

 

Lemma It follows that all C A for which k A C k lie in GA for k A k

To resume the pro of of Theorem given A A wenow dene a function f C A by

in the denition f z z A Since k f z f z k we see that f is continuous take

of continuity Because GAisopeninA by Lemma it follows from the top ological denition

 

of a continuous function that f GA is op en in A But f GA is the set of all z C where



z A has an inverse so that f GA A This set being op en its complement A is

closed



Finally dene g A A by g z z A For xed z A cho ose z C such that



 

jz z j k A z k From the pro of of Lemma with A A z and C A z

  

we see that z A as k A z A z k jz z j Moreover the p ower series

 

n

X

z z



z A z A

 

k 

converges for n by Lemma b ecause

 

k z z z A k jz z j k z A k

   

By Lemma the limit n of this p ower series is



X

z z z z

 

g z

z A z A z A z A z A

   

k 

Hence

X

k k 

g z z z z A

 

k 

ELEMENTARY THEORYOF C ALGEBRAS

 

is a normconvergentpower series in z For z we write k g z k jz j k I Az k and



observe that lim I Az I since lim k Az kby Hence lim I Az

z z z

Iand

lim k g z k

z

Let A b e a functional on A since is b ounded implies that the function g z g z

is given byaconvergentpower series and implies that

lim g z

z

Now supp ose that A so that AC The function g and hence g is then dened on

C where it is analytic and vanishes at innity In particular g is b ounded so that by Liouvilles

theorem it must be constant By this constant is zero so that g by Corollary

This is absurd so that A C hence A

The fact that the sp ectrum is never empty leads to the following GelfandMazur theorem

utative C algebras which will b e essential in the characterization of comm

Corollary If every element except of a unital Banach algebra A is invertible then A C

as Banach algebras

Since A for each A there is a z C for which A z I is not invertible Hence

A A

A z I by assumption and the map A z is the desired algebra isomorphism Since

A A

k A kk z I k jz j this isomorphism is isometric

Dene the sp ectral radius r AofA A by

r Asupfjz jz Ag

From Theorem one immediately infers

r A k A k

Prop osition For each A in a unital Banach algebra one has

n n

r A lim k A k

n

By Lemma for jz j k A k the function g in the pro of of Lemma has the norm

convergentpower series expansion

k

X

A

g z

z z

k 

On the other hand wehave seen that for any z A one maynda z A such that the



power series converges If jz j rAthenz A so converges for jz j rA At

this p oint the pro of relies on the theory of analytic functions with values in a Banach space which

says that accordingly is normconvergentfor jz j rA uniformly in z Comparing with

this sharp ens what weknow from Lemma The same theory says that cannot

n n

normconverge uniformly in z unless k A k jz j for large enough n This is true for all z for

which jz j rA so that

n

lim sup k A k r A

n

To derive a second inequalitywe use the following p olynomial sp ectral mapping prop erty

Lemma Forapolynomial p on C dene p A as fpz j z Ag Then

p A pA

Banach algebra basics

To prove this equalitycho ose z C and compare the factorizations

n

Y

pz c z

i

i

n

Y

pA I c A I

i

i

Here the co ecients c and are determined by p and When pA then pA I is

i

invertible which implies that all A I must b e invertible Hence pA implies that

i

at least one of the A I is not invertible so that A for at least one i Hence

i i

p ie p A This proves the inclusion pA p A

i

z for some z A so that for some i one must Conversely when p A then p

have z for this particular z Hence A so that A is not invertible

i i i

implying that pA I is not invertible so that p A This shows that p A pA

and follows

To conclude the pro of of Prop osition we note that since A is closed there is an A

n n n n

for which jj r A Since A by Lemma one has j j k A k by Hence

n n

k A k jj r A Combining this with yields

n n n

lim sup k A k r A k A k

n

Hence the limit must exist and

n n n

lim k A k inf k A k r A

n n

Denition An ideal in a Banach algebra A is a closed linear subspace I A such that

A I implies AB I and BA I for al l B A

A leftideal of A is a closedlinear subspace I for which A I implies BA I for al l B A

A rightideal of A is a closed linear subspace I for which A I implies AB I for al l B A

A maximal ideal is an ideal I A for which no ideal I A I I exists which contains I

In particular an ideal is itself a Banach algebra An ideal I that contains an invertible element



A must coincide with A since A A I must lie in I so that all B B I must lie in I This

shows the need for considering Banach algebras with and without unit it is usually harmless to

add aunit to a Banach algebra A but a given prop er ideal I A do es not contain I and one

cannot add I to I without ruining the prop erty that it is a prop er ideal

Prop osition If I is an ideal in a Banach algebra A then the quotient AI is a Banach

algebra in the norm

k A k inf k A J k

J I

and the multiplication

A B AB

Here A AI is the canonical projection If A is unital then AI is unital with unit I

We omit the standard pro of that AI is a Banach space in the norm As far as the

Banach algebra structure is concerned rst note that is well dened when J J I one



has

A J B J AB AJ J B J J AB A B

  

since AJ J B J J I by denition of an ideal and J for all J I Toprove

 

observethat by denition of the inmum for given A A for each there exists a J I

such that

k A k k A J k

ELEMENTARY THEORYOF C ALGEBRAS

For if suchaJ would not exist the norm in AI could not b e given by On the other hand

for any J I it is clear from that

k A kk A J kk A J k

For A B A cho ose andJ J I such that holds for A B and estimate



k A B k k A J B J kk A J B J k

 

k A J B J kk A J kk B J k

 

k A k k B k

Letting yields k A B kk A kk B k

When A has a unit it is obvious from that I is a unit in AI By with A I

kk I k On the other hand from with B I one derives k I k one has k I

Hence k I k

Commutative Banach algebras

Wenow assume that the Banach algebra A is commutative that is AB BA for all A B A

Denition The structure space A of a commutative Banach algebra A is the set of

al l nonzero linear maps A C for which

AB A B

for al l A B A We say that such an is multiplicative

In other words A consists of al l nonzero homomorphisms from A to C

Prop osition Let A have a unit I

Each A satises

I

each A is continuous with norm

k k

hence

j Aj k A k

for al l A A

The rst claim is obvious since IA I A A and there is an A for which

A b ecause is not identically zero

For the second weknow from Lemma that A z is invertible when jz j k A k so that

A z A z since is a homomorphism Hence j Aj jz j for jz j A k and

follows

Theorem Let A be a unital commutative Banach algebra There is a bijective correspon

set of al l maximal ideals in A in that the kernel ker of each dence between A and the

A is a maximal ideal I each maximal ideal is the kernel of some Aand



I i I

The kernel of each A is closed since is continuous by Furthermore ker

is an ideal since satises The kernel of every linear map VC on a vector space V

has co dimension one that is dim V ker so that ker isamaximalideal Again on

ector space when ker ker then is a multiple of For A this implies anyv

  i

b ecause of



Commutative Banach algebras

Wenow show that every maximal ideal I of A is the kernel of some A Since I A

there is a nonzero B A which is not in I Form

I fBA J jA AJ Ig

B

This is clearly a leftideal since A is commutative I is even an ideal Taking A we see

B

I I Taking A I and J we see that B I so that I I Hence I A as I is

B B B B

maximal In particular I I hence I BA J for suitable A AJ I Apply the canonical

B

pro jection A AI to this equation giving

I I BA B A



b ecause of and J hence A B in AI Since B was arbitrary though

By Corollary this yields nonzero this shows that every nonzero elementofAI is invertible

AI C so that there is a homomorphism AI C Now dene a map A C by

A A This map is clearly linear since and are Also

A B A B A B AB AB

b ecause of and the fact that is a homomorphism

Therefore is multiplicative it is nonzero b ecause B or b ecause I Hence

A Finally I ker since I ker but if B I we saw that B so that

actually I ker

By wehaveA A Recall that the weak top ology also called w top ology

on the dual B of a Banach space B is dened by the convergence i v v for all

n n

v B The Gelfand top ology on A is the relative w top ology

Prop osition The structurespace A of a unital commutative Banach algebra A is com

pact and Hausdor in the Gelfand topology

The convergence in the w top ology by denition means that A A for all

n n

A A When A for all n one has

n

j AB A B j j AB AB A B A B j

n n n

j AB AB j j A B A B j

n n n

In the second term we write

B B A B A B A A B A

n n n n n

By and the triangle inequality the absolute value of the righthand side is b ounded by

k B k j A Aj k A k j B B j

n n

All in all when in the w top ology weobtainj AB A B j so that the limit

n

A Hence Aisw closed

the unit ball in A consisting of all functionals with norm From wehaveA A



By the BanachAlaoglu theorem the unit ball in A is w compact Being a closed subset

of this unit ball Ais w compact Since the w top ology is Hausdor as is immediate from

its denition the claim follows

Weembed A in A by A A where

A A

When A this denes A as a function on A By elementary functional analysis the w

top ology on A is the weakest top ology for whichallA A A are continuous This implies that

ELEMENTARY THEORYOF C ALGEBRAS

the Gelfand top ology on Aistheweakest top ology for which all functions A are continuous

In particular a basis for this top ology is formed by all op en sets of the form



A O f Aj A Og

where A A and O is an op en set in C

Seen as a map from A to C A the map A A dened by is called the Gelfand

transform

For any compact Hausdor space X we regard the space C X of all continuous functions on

X as a Banach space in the supnorm dened by

k f k sup jf xj

xX

A basic fact of top ology and analysis is that C X is complete in this norm Convergence in the

supnorm is the same as uniform convergence Whats more it is easily veried that C X iseven

acommutativeBanach algebra under p ointwise addition and multiplication that is

f g x f xg x

fgx f xg x

Hence the function which is for every x is the unit I One checks that the sp ectrum of

X

f C X is simply the set of values of f

We regard C A as a commutative Banac h algebra in the manner explained

Theorem Let A be a unital commutative Banach algebra

The Gelfand transform is a homomorphism from A to C A

The image of A under the Gelfand transform separates points in A

The spectrum of A A is the set of values of A on A in other words

A A fA j Ag

The Gelfand transform is a contraction that is

k A k k A k

The rst prop erty immediately follows from and When there is an A A



for which A A so that A A This proves

 



If A GA ie A is invertib e then A A so that A for all A

When A GA the ideal I fAB jB Ag do es not contain I so that it is contained in

A

a maximal ideal I this conclusion is actually nontrivial relying on the axiom of choice in the

guise of Hausdor s maximality priciple Hence by Theorem there is a A for which

A All in all wehaveshowed that A GA is equivalentto A forall A

Hence A z GAi A z for all A Thus the resolventis

Afz C j z A Ag

Taking the complement and using w e obtain

Eq then follows from and

Wenow lo ok at an example which is included for three reasons rstly it provides a concrete

illustration of the Gelfand transform secondly it concerns a commutative Banach algebra whichis

not a C algebra and thirdly the Banach algebra in question has no unit so the example illustrates

what happ ens to the structure theory in the absence of a unit In this connection let us note in

general that each A has a suitable extension to A namely

I

A I A

Commutative Banach algebras

The p oint is that remains multiplicative on A as can be seen from and the denition

I

This extension is clearly unique Even if one do es not actually extend A to A the existence

I

of shows that satises since this prop erty whichwas proved for the unital case holds

for and therefore certainly for the restriction of to A



Consider A L R with the usual linear structure and norm

Z

k f k dx jf xj



R

The asso ciative pro duct dening the Banach algebra structure is convolution that is

Z

dy f x y g y f g x

R

Strictly sp eaking this should rst b e dened on the dense subspace C R and subsequently b e

c



extended by continuityto L R using the inequality b elow Indeed using Fubinis theorem on

pro duct integrals we estimate

Z Z Z Z

k f g k dx j dy f x y g y j dy jg y j dx jf x y j



R R R R

Z Z

dy jg y j dx jf xj k f k k g k

 

R R

which is



There is no unit in L R since from one sees that the unit should be Diracs delta

function ie the measure on R which assigns to x andtoallotherx which do es not lie



in L R



We know from the discussion following that every multiplicative functional L R



is continuous Standard Banach space theory says that the dual of L R is L R Hence for



each L R there is a function L R suchthat

Z

f dx f x x

R

The multiplicativity condition then implies that x y x y for almost all

x y R This implies

x expipx

for some p C and since is b ounded b eing in L R it must b e that p R The functional

corresp onding to is simply called p It is clear that dierent ps yield dierent functionals



so that L R may b e identied with R With this notation we see from and that

the Gelfand transform reads

Z

ipx

dx f xe f p

R

Hence the Gelfand transform is nothing but the Fourier transform more generallymany of the

integral transforms of classical analysis maybe seen as sp ecial cases of the Gelfand transform

The wellknown fact that the Fourier transform maps the convolution pro duct into the

pointwise pro duct is then a restatement of Theorem Moreover we see from that



the sp ectrum f off in L R is just the set of values of its Fourier transform

Note that the Gelfand transform is strictly a contraction ie there is no equality in the b ound



Finally the RiemannLeb esgue lemma states that f L R implies f C R which



is the space of continuous functions on R that go to zero when jxj This is an imp ortant

function space whose denition may b e generalized as follows

Denition Let X be a Hausdor space X which is lo cally compact in that each point

has a compact neighbourhood The space C X consists of al l continuous functions on X which



vanish at innity in the sense that for each there is a compact subset K X such that

jf xj for al l x outside K

ELEMENTARY THEORYOF C ALGEBRAS

So when X is compact one trivially has C X C X When X is not compact the supnorm



can still b e dened and just as for C X one easily checks that C X is a Banach algebra



in this norm



We see that in the example A L R the Gelfand transform takes values in C A This



may b e generalized to arbitrary commutative nonunital Banach algebras The nonunital version

of Theorem is

Theorem Let A be a nonunital commutative Banach algebra

The structurespace A is local ly compact and Hausdor in the Gelfand topology

A The space A is the onepoint compactication of

I

The Gelfand transform is a homomorphism from A to C A



The spectrum of A A is the set of values of A on A with zeroadded if is not already

contained in this set

The claims and in Theorem hold

Recall that the onep oint compactication X of a noncompact top ological space X is the

set X whose op en sets are the op en sets in X plus those subsets of X whose complement

is compact in X If on the other hand X is a compact Hausdor space the removal of some p oint

yields a lo cally compact Hausdor space X X nfg in the relative top ology ie the op en

sets in X are the op en sets in X minus the p oint whose onep oint compactication is in turn

X

To prove weaddaunittoA and note that

A A

I

where each A is seen as a functional on A by and the functional is dened by

I

A I

There can b e no other elements of A b ecause the restriction of has a unique multiplicative

I

extension to A unless it identically vanishes on A In the latter case is clearly

I

the only multiplicative p ossibility

By Prop osition the space A is compact and Hausdor by one has

I

AA nfg

I

as a set In view of the paragraph following in order to prove and we need to show

that the Gelfand top ology of A restricted to A coincides with the Gelfand top ology of

I

A itself Firstly it is clear from that any op en set in Ainitsown Gelfand top ology

is the restriction of some op en set in A b ecause A A Secondly for any A A C and

I I

op en set OC from we evidently have

f A j A I Ognfg f Aj A Og

I

When do es not lie in the set fg on the lefthand side one should here omit the nfg

With this shows that the restriction of anyopen set in A toAisalways op en in

I

the Gelfand top ology of A This establishes and

It follows from and that

A

for all A A whichbycontinuityofA leads to

The comment preceding Theorem implies The nal claim follows from the fact

that it holds for A I

Commutative C algebras

Commutative C algebras

The BanachalgebraC X considered in the previous section is more than a Banach algebra Recall

Denition The map f f where

f xf x

evidently denes an involution on C X in which C X is a commutative C algebra with unit

The main goal of this section is to prove the converse statement cf Denition

Theorem Let A be a commutative C algebra with unit Then there is a compact Haus

dor space X such that A is isometrical ly isomorphic to C X This space is unique up to

homeomorphism

The isomorphism in question is the Gelfand transform so that X A equipp ed with the

Gelfand top ology and the isomorphism A C X isgiven by

AA

Wehave already seen in that this transform is a homomorphism so that is satised

Toshow that holds as well it suces to show that a selfadjointelementofA is mapp ed

into a realvalued function b ecause of and the fact that the Gelfand transform is

complexlinear

Wepick A A and A and supp ose that A i where R By

R

one has B i where B A I is selfadjoint Hence for t R one computes

j B itIj t t

On the other hand using and we estimate

j B itIj k B itI k k B itI B itI kk B t kk B k t

Using then yields t k B k for all t R For this is imp ossible For

we rep eat the argumentwith B B nding the same absurdity Hence sothat A

is real when A A Consequentlyby the function A is realvalued and follows as

announced

e that the Gelfand transform and therefore the morphism in is isometric Wenow prov

m m

When A A the axiom reads k A kk A k This implies that k A kk A k for all

m

m N Taking the limit in along the subsequence n then yields

r Ak A k

In view of and this implies

k A k k A k

For general A A we note that A A is selfadjoint so that wemay use the previous result and

to compute

d

A k k A k k A k k A A kk A A k k A

c

In the third equalitywe used A A whichwe just proved and in the fourth we exploited the

fact that C X isaC algebra so that is satised in it Hence holds for all A A

It follows that in is injective b ecause if A for some A then would fail

to b e an isometry A commutative Banach algebra for which the Gelfand transform is injective

is called semisimple Thus commutative C algebraa are semisimple

We nally prove that the morphism is surjective We know from that the image

C A b ecause A is closed b eing a C algebra hence a Banach space AA is closed in

In addition we know from that A separates p oints on A Thirdly since the Gelfand

transform was just shown to preserve the adjoint A is closed under complex conjugation by

Finally since I by and the image A contains The surjectivity

X X

of now follows from the following StoneWeierstrass theoremwhichwe state without pro of

ELEMENTARY THEORYOF C ALGEBRAS

Lemma Let X be a compact Hausdor space and regard C X as a commutative C

algebra as explained above A C subalgebra of C X which separates points on X and contains

coincides with C X

X

Being injective and surjective the morphism is bijective and is therefore an isomorphism

The uniqueness of X is the a consequence of the following result

d C X as a commutative C Prop osition Let X bea compact Hausdor space and regar

algebra as explainedabove Then C X equipped with the Gelfand topologyishomeomorphic

to X

Each x X denes a linear map C X C by f f x which is clearly multiplica

x x

tive and nonzero Hence x denes a map E for Evaluation from X to C X given

x

by

E xf f x

Since a compact Hausdor space is normal Urysohns lemma says that C X separates p oints on

X ie for all x y there is an f C X for which f x f y This shows that E is injective

We now use the compactness of X and Theorem to prove that E is surjective The

maximal ideal I I in C X which corresp onds to C X is obviously

x x

x

I ff C X j f xg

x

Therefore when E is not surjective there exists a maximal ideal I C X which for each x

X contains at a function f for which f x if not I would contain an ideal I which

x x x

thereby would not be maximal For each x the set O where f is nonzero is op en b ecause

x x

f is continuous This gives a covering fO g of X By compactness there exists a nite

x xX

P

N

jf j This function is strictly vering fO g Then form the function g sub co

x x iN

i i

i

p ositiveby construction so that it is invertible note that f C X isinvertible i f x for



all x X inwhich case f xf x But I is an ideal so that with all f I since all

x

i

f I also g I But an ideal containing an invertible elementmust coincide with A see the

x

comment after contradicting the assumption that I is a maximal ideal

Hence E is surjective since w e already found it is injective E must b e a bijection It remains

to b e shown that E is a homeomorphism Let X denote X with its originally given top ology and

o



write X for X with the top ology induced by E Since f E f by and and the

G

Gelfand top ology on C X is the weakest top ology for which all functions f are continuous

we infer that X is weaker than X since f lying in C X is continuous Here a top ology T

G o o 

is called weaker than a top ology T on the same set if any op en set of T containsanopensetof



T This includes the p ossibility T T



Without pro of wenow state a result from top ology

Lemma L et a set X be Hausdor in some topology T and compact in a topology T If T

 

is weaker than T then T T



Since X and X are b oth compact and Hausdor the former by assumption and the lat

o G

ter by Prop osition we conclude from this lemma that X X in other words E is a

 G

homeomorphism This concludes the pro of of

Prop osition shows that X as a top ological space may be extracted from the Banach

up to homeomorphism Hence if C X C Y as a C algebra algebraic structure of C X

where Y is a second compact Hausdor space then X Y as top ological spaces Given the

isomorphism A C X constructed ab ove a second isomorphism A C Y is therefore only

p ossible if X Y This proves the nal claim of Theorem

The condition that a compact top ological space b e Hausdor is sucient but not necessary

et C X is for the completeness of C X in the supnorm However when X is not Hausdor y

complete the map E may fail to b e injective since in that case C X may fail to separate p oints

on X

Commutative C algebras

On the other hand supp ose X is lo cally compact but not compact and consider A C X

b

this is the space of all continuous b ounded functions on X Equipp ed with the op erations

and this is a commutative C algebra The map E X C X is now injective

b

but fails to b e surjective this is suggested by the invalidityoftheproofwegaveforC X Indeed

it can b e shown that C X is homeomorphic to the CehStone compactication of X

b

Let us now consider what happ ens to Theorem when A has no unit Following the strategy

weusedinproving Theorem wewould like to add a unit to A As in the case of a general

Banach algebra cf section weformA by dene multiplication by and use the

I

natural involution

A I A I

However the straightforward norm cannot b e used since it is not a C norm in that axiom

is not satised Recall Denition

Lemma Let A beaC algebra

The map A BA given by

AB AB

establishes an isomorphism between A and A BA

When A has no unit dene a norm on A by

I

k A I kk AI k

where the norm on the righthand side is the operator norm in B A and I on the

righthand side is the unit operator in BA With the operations and the

norm turns A into a C algebra with unit

I

By wehave k AB kk AB kk A kk B k for all B sothatk A kk A k by

On the other hand using and we can write

A

k A k k A kk AA k k A kk A

k A k

in the last step we used and k A k A k k Hence

k A kk A k

Being isometric the map must b e injective it is clearly a homomorphism so that wehave proved

It is clear from and that the map A I AI where the symbol I on the

lefthand side is dened below and the I on the righthand side is the unit in BA is a

morphism Hence the norm satises b ecause is satised in BA Moreover in

order to prove that the norm satises by Lemma it suces to prove that

k AI k k AI AI k

for all A A and C Todosoweuseatrick similar to the one involving but with inf

replaced by sup Namely in view of for given A BB and there exists a v Vwith

k v ksuch that k A k k Av k Applying this with BA and A AIwe infer

that for every there exists a B A with norm suchthat

k AI k k AIB k k AB B k k AB B AB B k

Here we used in A Using the righthand side may b e rearranged as

k B A IA IB kk B kk AI AI kk B k

Since k B kk B kk B k by and and k B k also in the last term the

inequality follows by letting

Hence the C algebraic version of Theorem is

ELEMENTARY THEORYOF C ALGEBRAS

Prop osition For every C algebra without unit there exists a unique unital C algebra A

I

and an isometric hence injective morphism A A such that A A C

I I

The uniqueness of A follows from Corollary b elow On the other hand in view of the

I

fact that b oth and dene a norm on A satisfying the claims of Prop osition we

I

conclude that the unital Banach algebra A called for in that prop osition is not in general unique

I

In any case having established the existence of the unitization of an arbitrary nonunital C

algebra we see that in particular a commutative nonunital C algebra has a unitization The

passage from Theorem to Theorem may then b e rep eated in the C algebraic setting

the only nontrivial p oint compared to the situation for Banach algebras is the generalization of

Lemma This now reads

Lemma L et X bea local ly compact Hausdor space and regard C X as a commutative



C algebra as explainedbelow Denition

A C subalgebra A of C X which separates points on X and is such that for each x X



thereisanf A such that f x coincides with C X



At the end of the daywe then nd

Theorem Let A be a commutative C algebra without unit There is a local ly compact

Hausdor space X such that A is isometrical ly isomorphic to C X This space is unique up to



homeomorphism

Sp ectrum and functional calculus

We return to the general case in whichaC algebra A is not necessarily commutative but assumed

unital but analyze prop erties of A by studying certain commutative subalgebras This will lead

to imp ortant results

For each element A A there is a smallest C subalgebra C A Iof A which contains A

and I namely the closure of the linear span of I and all op erators of the typ e A A where

 n

A is A or A Following the terminology for op erators on a Hilb ert space an element A A is

i

called normal when A A The crucial prop ertyof a normal op erator is that C A Iis

commutative In particular when A is selfadjoint C A I is simply the closure of the space of

all p olynomials in A It is sucient for our purp oses to restrict ourselves to this case

Theorem Let A A be a selfadjoint element of a unital C algebra

The spectrum A of A in A coincides with the spectrum A of A in C A I so

A

C AI

that we may unambiguously speak of the spectrum A

The spectrum A isasubsetof R

The structurespace C A I is homeomorphic with A so that C A I is isomorphic

to C A Under this isomorphism the Gelfand transform A A R is the identity

function id t t

A



GA be normal in A and consider the C algebra C A A I Recall Let A

    

generated by A A andI One has A A andA A A A and I all commute



with each other Hence C A A Iiscommutative it is the closure of the space of all p olynomials

  

in A A A A andI By Theorem wehave C A A I C X for some compact

Hausdor space X Since A is invertible and the Gelfand transform is an isomorphism A

is invertible in C X ie Ax x for all x X However for any f C X that is nonzero

throughout X wehave k f k ff p ointwise so that kf k ff p ointwise

X

hence

k ff k f k k

X

Here f is given by Using Lemma in terms of I wemay therefore write

X

k

X

f ff

I

f k f k k f k

k 

Sp ectrum and functional calculus



Hence A is a normconvergent limit of a sequence of p olynomials in A and A Gelfand trans

 

forming this result backto C A A I we infer that A is a normconvergent limit of a sequence

 

of p olynomials in A and A Hence A lies in C A I and C A A I C A I

Now replace A by A z where z C When A is normal A z is normal So if we assume that

A z GA the argumentabove applies leading to the conclusion that the resolvent Ain

A

A coincides with the resolvent AinC A I By Denition we then conclude that

C AI

A A

A

C AI

According to Theorem the function A is realvalued when A A Hence by the

sp ectrum A is real so that by the previous result A is real

C AI

Finallygiven the isomorphism C A I C X of Theorem where X C A I

according to the function A is a surjectiv emapfrom X to A Wenowprove injectivity

When X and A A then for all n N wehave

 

n n n n

A A A A

 

by iterating with B A Since also I I by we conclude by linearity



that on all p olynomials in A By continuity cf this implies that on

 

C A I since the linear span of all p olynomials is dense in C A I Using wehave proved

that A A implies

 

Since A C X by A is continuous To provecontinuity of the inverse one checks

 n n

that for z A the functional A z C A I maps A to z and hence A to z etc



Lo oking at one then sees that A is continuous In conclusion A is a homeomorphism

The nal claim in is then obvious

An immediate consequence of this theorem is the continuous functional calculus

Corollary For each selfadjoint element A A and each f C A thereisanoperator

f A A which is the obvious expression when f is a polynomial and in general is given via the

uniform approximation of f by polynomials such that

f A f A

k f A k k f k

In particular the norm of f A in C A I coincides with its norm in A

Theorem yields an isomorphism C A C A I which is precisely the map f

f A of the continuous functional calculus The fact that this isomorphism is isometric see

yields Since f A is the set of values of f on A follows from with

A f A

The last claim follows bycombining with and

Corollary The norm in a C algebra is unique that is given a C algebra A ther e is no

other norm in which A is a C algebra

First assume A A and apply with f id By denition cf the supnorm

A

of id is r A so that

A

k A k r A A A

Since A A is selfadjointforany A for general A A wehave using

p

r A A k A k

Since the sp ectrum is determined by the algebraic structure alone shows that the norm is

determined by the algebraic structure as well

Note that Corollary do es not imply that a given algebra can b e normed only in one way

so as to b e completed into a C algebra we will in fact encounter an example of the opp osite

situation In the completeness of A is assumed from the outset

ELEMENTARY THEORYOF C ALGEBRAS

Corollary A morphism A B between two C algebras satises

k A kk A k

and is therefore automatical ly continuous



When z A so that A z exists in AthenA z is certainly invertible in B for

 

implies that A z A z Hence A A so that

A A

Hence r A r A so that follows from

For later use we note

Lemma When A B is a morphism and A A then

f A f A

for al l f C A here f A is dened by the continuous functional calculus and so is f A

in view of

The prop ertyis true for p olynomials by since for those f has its naive meaning For

general f the result then follows bycontinuity

Positivit y in C algebras

A b ounded op erator A BH on a Hilb ert space H is called p ositive when A for



all H this prop erty is equivalent to A A and A R and clearly also applies to

closed subalgebras of BH In quantum mechanics this means that the exp ectation value of the

observable A is always p ositive

Classically a function f on some space X is p ositive simply when f x for all x X This

of the commutative C algebra C X where X is a lo cally applies in particular to elements



compact Hausdor space Hence wehave a notion of p ositivity for certain concrete C algebras

whichwewould like to generalize to arbitrary abstract C algebras Positivity is one of the most

imp ortant features in a C algebra it will for example play a central role in the pro of of the

Gelfand Neumark theorem In particular one is interested in nding a number of equivalent

characterizations of p ositivity

Denition An element A of a C algebra A is cal led p ositive when A A and its spec

 

trum is positive ie A R We write A or A A where

 

A fA A j A R g

R

It is immediate from Theorems and that A A is p ositive i its Gelfand

R

transform A is p ointwise p ositivein C A



Prop osition The set A of al l positive elements of a C algebra A is a convex cone that

is

  

when A A and t R then tA A

 

when A B A then A B A

 

A A

Positivityin C algebras

The rst prop erty follows from tAt A which is a sp ecial case of

Since A rA we have jc tj c for all t A and all c r A Hence

sup jc A j c by and so that k c A k c Gelfand transform

A A

t A

A I this implies k cI A k c for all c k A k by Inverting this argument ing backtoC



one sees that if k cI A k c for some c k A k then A R

Use this with A A B and c k A k k B k clearly c k A B k by Then

k cI A B kk k A kA k k k B kB k c

where in the last step we used the previous paragraph for A and for B separately As we have



seen this inequality implies A B A

 

Finally when A A and A A it must b e that A hence A by and

This is imp ortant b ecause a convex cone in a real vector space is equivalenttoalinear partial

ordering ie a partial ordering in which A B implies A C B C for all C and A B



for all R The real vector space in question is the space A of all selfadjoint elements of A

R





The equivalence b etween these two structures is as follows given A A one denes A B if

R

 

B A A and given one puts A fA A j Ag

R

R R

For example when A A one checks the validityof

k A k I A k A k I

by taking the Gelfand transform of C A I The implication

B A B k A kk B k

then follows b ecause B A B and for A B yield kB k I A k B k I so that

A k B k k B k hence k A kk B k by and For later use we also record



Lemma When A B A and k A B k k then k A k k

By wehave A B k I hence A k I B by the linearity of the partial ordering

obtain which also implies that k I B k Ias B Hence using k I since k we

k I A k I from which the lemma follows by

Wenow come to the central result in the theory of p ositivityin C algebras which generalizes

the cases A BH and A C X



Theorem One has



A fA j A A g

R

fB B j B Ag

p



A A is dened by the continuous functional calculus When A R and A A then

R

p

p



for f and satises A A Hence A fA j A A g The opp osite inclusion follows

R

from and This proves



The inclusion A fB B j B Ag is is trivial from

Lemma Every selfadjoint element A has a decomposition A A A where A A

 



A and A A Moreover k A kk A k



t f t Apply the continuous functional calculus with f id f f where id

 

A A

maxft g and f t maxft g Since k f k r A k A k where we used the

b ound follows from with A A



We use this lemma to prove that fB B j B Ag A Apply the lemma to A B B noting

that A A Then

A A A A BA BA BA A A AA A B





Since A R b ecause A is p ositive we see from with f t t that A

Hence BA BA

ELEMENTARY THEORYOF C ALGEBRAS



Lemma If C C A for some C A then C

By we can write C D iE D E A so that

R

C C D E CC

Nowforany A B A one has

AB fg BA fg

This is b ecause for z the invertibilityof AB z implies the invertibilityof BA z Namely one

  

computes that BA z B AB z A z I Applying this with A C and B C we



see that the assumption C C R implies CC R hence CC R By

 

and we see that C C ie C C R so that the assumption C C A

now yields C C Hence C by

A BA BA The last claim b efore the lemma therefore implies BA As



we see that A and nally A by the continuous functional calculus with f tt



Hence B B A which lies in A



An imp ortant consequence of is the fact that inequalities of the typ e A A for



A A A are stable under conjugation by arbitrary elements B AsothatA A implies

 R 

B A B B A B This is b ecause A A is the same as A A by there is an A A

  

such that A A A A But clearly A B A B and this is nothing but B AB B A B



For example replace A in by A A and use yielding A A k A k I Applying the

ab ove principle gives

B A AB k A k B B

for all A B A

Ideals in C algebras

An ideal I in a C algebra A is dened by As wehave seen a prop er ideal cannot contain

I in order to prove prop erties of ideals we need a suitable replacement of a unit

Denition An approximate unit in a nonunital C algebra A is a family fI g where



is some directed set ie a set with a partial order and a sense in which with the

fol lowing properties

I I

and I sothat

k I k

k I A A k lim k AI A k lim

for al l A A

For example the C algebra C R has no unit the unit would b e which do es not vanish

 R

at innity b ecause it is constant but an approximate unit maybe constructed as follows take

N and take I to b e a continuous function which is on n nand v anishes for jxj n

n

One checks the axioms and notes that one certainly do es not have I in the supnorm

n R

Prop osition Every nonunital C algebra A has an approximate unit When A is separable

ontaining a countable dense subset then may be taken to becountable in c

Ideals in C algebras

One takes to b e the set of all nite subsets of A partially ordered by inclusion Hence

P

is of the form fA A g from whichwe build the element B A A Clearly B is

 n i

i

i

 

selfadjoint and according to and one has B R sothatn I B is invertible

in A Hence wemayform

I

 

I B n I B

 

Since B is selfadjointandB commutes with functions of itself suchasn I B one has

 

I I Although n I B is computed in A sothatitisoftheformC I for some C A

I

and C one has I B C B which lies in A Using the continuous functional calculus on

B with f ttn t one sees from and the p ositivityofB that I

Putting C I A A a simple computation shows that

i i i

X



C n B n I B C

i

i

i



Wenow apply with A B and f tn tn t Since f and f assumes its



maximum at t n one has sup jf tj n As B R it follows that k f k n

tR

P



hence k n B n I B k n by so that k C C k n by Lemma

i

i

i

then shows that k C C k n for each i n Since any A A sits in some directed

i

i

subset of with n itfollows from that

lim k I A A k lim k I A A I A A k lim k C C k

i

i

The other equality in follows analogously

Finally when A is separable one may draw all A o ccurring as elements of from a

i

countable dense subset so that is countable

The main prop erties of ideals in C algebras are as follows

Theorem Let I beanideal in a C algebra A

If A I then A I in other words every ideal in a C algebra is selfadjoint

The quotient AI is a C algebra in the norm the multiplication and the

involution

A A

Note that is well dened b ecause of

Put I fA j A Ig Note that J I implies J J I I it lies in I b ecause I is an ideal

hence a leftideal and it lies in I because I is an ideal hence a rightideal Since I is an ideal

I I is a C subalgebra of A Hence by it has an approximate unit fI g Take J I Using

and we estimate

J J I kk JJ JJ I I JJ JJ I k k J J I k k J I J

k JJ JJ I k k I JJ JJ I kk J J J J I k k I kk JJ JJ I k

As wehaveseen J J I I so that also using b oth terms vanish for Hence

lim k J J I k But I lies in I I so certainly I I and since I is an ideal it must

b e that J I I for all Hence J is a normlimit of elements in IsinceI is closed it follows

that J I This proves

In view of all we need to prove to establish is the prop erty This uses

Lemma Let fI g be an approximate unit in I and let A A Then

k k A k lim k A AI

ELEMENTARY THEORYOF C ALGEBRAS

It is obvious from that

k A AI kk A k

To derive the opp osite inequalityadda unit I to A if necessary pickany J I and write

k A AI kk A J I I J I I kk A J kk I I k k J I J k

Note that

k I I k

by and the pro of of The second term on the righthand side go es to zero for

since J I Hence

lim k A AI kk A J k

For each wecancho ose J I so that holds For this sp ecic J we combine

and to nd

lim k A AI k k A kk A AI k

Letting proves

Wenowprove in AI Successively using in A and

I

wend

k A k lim k A AI k lim k A AI A AI k

lim k I I A AI I k lim k I I kk A AI I k lim k A AI I k

k A A kk A A kk A A k

Lemma then implies

This seemingly technical result is very imp ortant

Corollary The kernel of a morphism between two C algebras is an ideal Conversely every

ideal in a C algebra is the kernel of some morphism Hence every morphism has norm

The rst claim is almost trivial since A implies AB BA for all B by

Also since is continuous see its kernel is closed

The converse follows from Theorem since I is the kernel of the canonical pro jection

A AI where AI is a C algebra and is a morphism by and

The nal claim follows from the preceding one since k k

For the next consequence of we need a

Lemma An injective morphism between C algebras is isometric In particular its range

is closed

Assume there is an B A for which k B k k B k By and this

By and we implies k B B k k B B k Put A B B noting that A A

must have A A Then implies A A By Urysohns lemma there is a

nonzero f C A whichvanishes on A so that f A By Lemma wehave

f A contradicting the injectivityof

Corollary The image of a morphism A B between two C algebras is closed In

particular A is a C subalgebraof B

Dene A ker B by AA where A is the equivalence class in A ker

of A A By the theory of vector spaces is a vector space isomorphism between A ker

and A and In particular is injective According to and the space

and are morphisms is a C algebra morphism Hence A ker is a C algebra Since

A ker has closed range in B by But A ker A so that has closed range

in B Since is a morphism its image is a algebra in B which by the preceding sentence is

closed in the norm of B Hence A inheriting all op erations in BisaC algebra

States

States

The notion of a state in the sense dened b elow comes from quantum mechanics but states play

acentral role in the general theory of abstract C algebras also

Denition A state on a unital C algebra A is a linear map A C which is p ositive



in that A for al l A A and normalized in that

I

The state space S A of A consists of al l states on A

For example when A BH then every H with norm denes a state by

AA

Positivityfollows from Theorem since B B k B k and normalization is obvious

from I

Theorem The state spaceof A C X consistsofallprobability measures on X

By the Riesz theorem of measure theoryeach p ositive linear map C X C is given bya

regular p ositive measure on X The normalization of implies that X so

X

that is a probability measure

The p ositivityof with implies that A B A B denes a preinner pro duct on

A Hence from we obtain

j A B j A A B B

which will often b e used Moreover for all A A one has

A A

as A A I A I IA A

Partly in order to extend the denition of a state to nonunital C algebras wehave

Prop osition A linear map A C on a unital C algebra is positive i is bounded

and

k k I

In particular

A state on a unital C algebraisbounded with norm

An element A for which k k I is a state on A

Aj I k A k For When is p ositive and A A we have using the b ound j

general A we use with A I and the b ound just derived to nd

j B j B B I I k B B k I k B k

Hence k k I Since the upp er b ound is reached by B Iwehave

To prove the converse claim we rst note that the argument around ma y be copied

showing that is real on A Next weshowthat A implies A Cho ose s small

R

enoughsothatk I sA k Then assuming

k k j I sAj

k I sA k k I sA k

I I

Hence j I s Aj I which is only p ossible when A

Wenow pass to states on C algebras without unit Firstlywe lo ok at a state in a more general

context

ELEMENTARY THEORYOF C ALGEBRAS

Denition A p ositivemapQ A B between two C algebras is a linear map with the

property that A implies QA in B

Prop osition Apositive map between two C algebras is boundedcontinuous



Let us rst show that b oundedness on A implies b oundedness on A Using and

we can write

A A A iA iA

 

where A etc are p ositive Since k A kk A k and k A kk A k by wehave k B kk A k





for B A A A or A by Hence if kQB k C k B k for all B A and some

 

C then

kQA kk QA k kQA k kQA k kQA k C k A k

 



so that Now assume that Q is not b ounded by the previous argumentitisnotboundedonA

 



for each n N there is an A A so that kQA k n here A consists of all A A with

n n

 

P



n A obviously converges to some A A Since Q is p ositive we k A k The series

n

n

have QA n QA for each n Hence by

n

kQA k n kQA k n

n

for all n N whichis imp ossible since kQA k is some nite number Thus Q is b ounded on



A and therefore on A by the previous paragraph

Cho osing B C we see that a state on a unital C algebra is a sp ecial case of a p ositive map

between C algebras Prop osition then provides an alternative pro of of Hence in the

nonunital case wemay replace the normalization condition in as follows

Denition A state on a C algebra A is a linear map A C which is positive and has

norm

This denition is p ossible by and is consistent with b ecause of The following

result is very useful cf

Prop osition A state on a C algebra without unit has a unique extension to a state

I

on the unitization A

I

The extension in question is dened by

A I A

I

This obviously satises it remains to prove p ositivity

Since a state on A is b ounded bywehave j A AI jforany approximate unit

in A The derivation of and may then b e copied from the unital case in particular

one still has j Aj A A Combining this with we obtain from that

A I A I j A j

I

Hence is p ositiveby

There are lots of states

Lemma For every A A and a A there is a state on A for which Aa When

a

A A there exists a state such that j Aj k A k

Representations and the GNSconstruction

If necessary weaddaunittoA this is justied by Dene a linear map C A CI C

a

by A I a Since a Aonehasa A I this easily follows from

a

the denition of Hence with A A I implies j A IjkA I k Since

a

I it follows that k k By the HahnBanach Theorem there exists an extension

a

of to A of norm By is a state which clearly satises A Aa

a a a a

Since A is closed by there is an a A for which r Ajaj For this a one has

j Aj jaj r Ak A k by

An imp ortant feature of a state space S A is that it is a convex set A convex set C in a

vector space C is a subset of V such that the conv ex sum v w b elongs to C whenever

P

v w C and Rep eating this pro cess it follows that p v b elongs to C when all

i i

i

P

p and p and all v C In the unital case it is clear that S Aisconvex since b oth

i i i

i

p ositivity and normalization are clearly preserved under convex sums In the nonunital case one

arrives at this conclusion most simply via

We return to the unital case Let S A b e the state space of a unital C algebra A Wesaw

in that each element of S Aiscontinuous so that S A A Since w limits obviously

preserve p ositivity and normalization we see that S A is closed in A if the latter is equipp ed

with the w top ology Moreover S A is a closed subset of the unit ball of A by so that

S A is compact in the relative w top ology by the BanachAlaoglu theorem

It follows that the state space of a unital C algebra is a compact convex set The very

simplest example is A C in whichcaseS Aisapoint

The next case is A C C C The dual is C as well so that each elementof C is

of the form c c Positive elements of C C are fo the form with

 

and so that a p ositive functional must have c and c Finally since I



normalization yields c c We conclude that S C C may b e identied with the interval



Now consider A M C We identify M C with its dual through the pairing A

Tr A It follows that S A consists of all p ositive matrices with Tr these are the

density matrices of quantum mechanics Toidentify S A with a familiar compact convex set we

parametrize

x y iz

y iz x

where x y z R The p ositivity of this matrix then corresp onds to the constraint x y z

Hence S M C is the unit ball in R

Representations and the GNSconstruction

The material of this section explains how the usual Hilb ert space framework of quantum mechanics

emerges from the C algebraic setting

Denition A representation of A on a Hilbert space H is a complex linear map

A BH satisfying

A B A B

A A

for al l A B A

A representation is automatically continuous satisfying the b ound

k A kk A k

This is b ecause is a morphism cf In particular k A kk A k when is faithful by

Lemma

There is a natural equivalence relation in the set of all representations of A two representa

tions on Hilb ert spaces H H resp ectively are called equivalent if there exists a unitary

 

isomorphism U H H suchthat U AU A for all A A

 

ELEMENTARY THEORYOF C ALGEBRAS

The map A for all A A is a representation more generallysuch trivial mayoccur

as a summand To exclude this p ossibility one says that a representation is nondegenerate if

is the only vector annihilated by all representatives of A

A representation is called cyclic if its carrier space H contains a cyclic vector for this

means that the closure of A whichinany case is a closed subspace of H coincides with H

Prop osition Any nondegenerate representation is a direct sum of cyclic representations

The pro of uses a lemma which app ears in many other pro ofs as well

Lemma Let M be a algebra in BH take a nonzer o vector H and let p be the

projection onto the closureof M Then p M that is p A for al l A M

If A M then ApH pH by denition of p Hence p Ap with p I p this reads

Ap pAp When A A then

Ap pA pAp pAp Ap

so that A p By this is true for all A M

lemma with M A the assumption of nondegeneracy guarantees that p is Apply this

nonzero and the conclusion implies that A p A denes a subrepresentation of A on pH This

subrepresentation is clearly cyclic with cyclic vector This pro cess may b e rep eated on p H

etc

If is a nondegenerate representation of A on H thenanyunitvector H denes a state

SA referred to as a vector state relativeto by means of Converselyfromany

state S A one can construct a cyclic representation on a Hilb ert space H with cyclic

ay We restrict ourselves to the unital case the general case follows vector in the following w

by adding a unit to A and using

Construction Given SA dene the sesquilinear form on A by



A B A B



Since is a state hencea positive functional this form is positive semidenite this means

that A A for al l A Its nul l space



N fA A j A Ag

isaclosed leftideal in A

The form projects to an inner product on the quotient AN If V A AN



is the canonical projection then by denition

VAVB A B



The Hilbert space H is the closureof AN in this inner product

The representation A is rstly denedonAN H by

AVB VAB

ontinuous it fol lows that is continuous Hence A may be dened on al l of H by c

extension of where it satises

The cyclic vector is denedby V Isothat

A A A A

The GelfandNeumark theorem

Wenowprovethevarious claims made here First note that the null space N of can b e



dened in two equivalentways

N fA A j A A g fA A j A B B Ag

 

The equivalence follows from the CauchySchwarz inequality The equality implies

that N is a leftideal which is closed b ecause of the continuityof This is imp ortant b ecause

it implies that the map A A A dened in quotients well to a map from AN to

AN the latter map is dened in Since is a morphism it is easily checked that

is a morphism as well satisfying on the dense subspace AN of H

To prove that is continuous on AN we compute k A k for VB where

and A B A By and step ab ove one has k A k B AA B By

the p ositivity of one has B AA B k A k B B But B B k k so that

k A kk A kk kuponwhich

k A kk A k

follows from

For later use we mention that the GNSconstruction yields

A B A B

Putting B A yields

k A k A A

whichmay alternatively b e derived from and the fact that is a representation

Prop osition If A H is cyclic then the GNSrepresentation A H denedbyany

vector state corresponding to a cyclic unit vector H is unitarily equivalent to A H

This is very simple to prove the op erator U H Himplementing the equivalence is initially

this op erator is welldened for dened on the dense subspace A by U A A

A implies A by the GNSconstruction It follows from that U is

unitary as a map from H to U H but since is cyclic for the image of U is H Hence U is

unitary It is trivial to verify that U intertwines and

Corollary If the Hilbert spaces H H of two cyclic representations each contain a

 

cyclic vector H H and

 

A A A A

   

for al l A A then A and A areequivalent



By the representation is equivalent to the GNSrepresentation and is equivalent



to On the other hand and are induced by the same state so they must coincide

The GelfandNeumark theorem

One of the main results in the theory of C algebras is

Theorem A C algebra is isomorphic to a subalgebraof BH for some Hilbert space H

The GNSconstruction leads to a simple pro of this theorem which uses the following notion

The universal representation of a C algebra A is the direct sum of al l Denition

u

its GNSrepresentations SAhence it is dened on the Hilbert space H H

u

S A

ELEMENTARY THEORYOF C ALGEBRAS

Theorem then follows by taking H H the desired isomorphism is Toprove that

u u

is injective supp ose that A for some A A By denition of a direct sum this implies

u u

A for all states Hence A hence k A k by this means

A A for all states which implies k A A k by Lemma so that k A k by

and nally A by the denition of a norm

Being injective the morphism is isometric by Lemma

u

While the universal representation leads to a nice pro of of the Hilb ert space H is

u

absurdly large in practical examples a b etter way of obtaining a faithful representation always

exists For example the b est faithful representation of BH is simply its dening one

Another consequence of the GNSconstruction or rather of is



Corollary An operator A A is positive that is A A i A for al l cyclic

R

representations

Complete p ositivity

Q cf Denition generalizes the notion of a state in that Wehave seen that a p ositivemap

the C in A C is replaced by a general C algebra B in Q A B Wewould liketoseeif

one can generalize the GNSconstruction It turns out that for this purp ose one needs to imp ose

a further condition on Q

n

We rst intro duce the C algebra M Aforagiven C algebra A and n N The elements

n

of M A are n n matrices with entries in A multiplication is done in the usual way ie

P

MN M N with the dierence that one now multiplies elements of A rather than

ij ik kj

k

n

complex numbers In particular the order has to b e taken into account The involution in M A

is of course given by M M in which the involution in A replaces the usual complex

ij

ji

n n

conjugation in C One may identify M Awith A M C in the obvious way

When is a faithful representation of A which exists by Theorem one obtains a faithful

n n

realization of M AonHC dened by linear extension of M v M v we here

n n i ij j

n

lo ok at elements of HC as ntuples v v where each v H The norm k M k of

 n i

n n

M M A is then simply dened to be the norm of M Since M A is a closed

n n

n n

algebra in BHC b ecause n it is obvious that M AisaC algebra in this norm

The norm is unique by Corollary so that this pro cedure do es not dep end on the choice of

Denition A linear map Q A B between C algebras is cal led completely p ositive

n n

if for al l n N the map Q M A M B denedbyQ M QM ispositive

n n ij ij

n

For example a morphism is a completely p ositive map since when A B B in M A then

n

tation of A on H is a A B B whichispositiveinM B In particular any represen

completely p ositive map from A to BH

If we also assume that A and B are unital and that Q is normalized we get an interesting

generalization of the GNSconstruction which is of central imp ortance for quantization theory

This generalization will app ear as the pro of of the following Stinespring theorem

Theorem Let Q A B be a completely positive map between C algebras with unit

such that QI I By Theorem we may assume that B is faithful ly repr esented as a

subalgebra B B BH for some Hilbert space H

There exists a Hilbert space H a representation of A on H and a partial isometry

W H H with W W I such that

QA W AW A A

Equivalently with p WW the target proje ction of W on H H pH H and U

H H denedas W seen as map not from H to H butasamapfrom H to H so that U

is unitary one has



U QAU p Ap

Complete p ositivity

The pro of consists of a mo dication of the GNSconstruction It uses the notion of a partial

isometry This is a linear map W H H between two Hilb ert spaces with the prop erty that



H contains a closed subspace K suchthatW W for all K andW

   

on K Hence W is unitary from K to W K It follows that WW K and W W K are

  



pro jections onto the image and the kernel of W resp ectively

We denote elements of H by v w with inner pro duct v w

Construction Dene the sesquilinear form on A H algebraic tensor prod



uct by sesquilinear extension of

A v B w v QA B w



Since Q is completely positive this form is positive semidenite denote its nul l space by

N

The form projects to an inner product on A H N If V A H A H N



is the canonical projection then by denition

V A v V B w A v B w



The Hilbert space H is the closureof A H N in this inner product

The representation A is initial ly denedonA H N by linear extension of

AV B w V AB w

N One has the bound this is wel ldened because AN

k A kk A k

so that A may bedenedonallof H by continuous extension of This extension

satises A A

The map W H H denedby

Wv V I v

isapartial isometry Its adjoint W H H is given by continuous extension of

W V A v QAv

from which the properties W W I and fol low

To show that the form dened by is p ositive wewrite

X X

v QA A v A v A v

i j j i i j j

i



ij ij

n

Now consider the element A of M A with matrix elements A A A Lo oking in a faithful

ij j

i

representation as explained ab ove one sees that

n

X X

A z A z k Az k z A z A z z A

i i j j j j i

i

ij ij

P

where Az A z Hence A Since Q is completely p ositive it must b e that B dened by

i i

i

n

its matrix elements B QA A is p ositivein M B Rep eating the ab ove argumentwith

ij j

i

A and replaced by B and resp ectively one concludes that the righthand side of is

p ositive

A B B A we To prove one uses in M A Namely for arbitrary

 n n

conjugate the inequality A AI k A k I with the matrix B whose rst rowisB B

n n  n

ELEMENTARY THEORYOF C ALGEBRAS

and which has zeros everywhere else the adjoint B is then the matrix whose rst column is

T

B B and all other entries zero This leads to B A AB k A k B B Since Q is

 n

completely p ositive one has Q B A AB k A k Q B B Hence in any representation B

n n

n

and anyvector v v H C one has

 n

X X

v QB A AB v k A k v QB B v

i j j i j j

i i

ij ij

P

V B v from and one then has With

i i

i

X X

k A k AB v AB v v QB A AB v

i i j j i j j

i



ij ij

X X

k A k v QB B v k A k B v B v

i j j i i j j

i



ij ij

X X

k A k V B v V B v k A k k k

i i j j

i j

To showthat W is a partial isometry use the denition to compute

WvWw V I v V I w I v I w v w



where we used and QI I

Tocheck one merely uses the denition of the adjoint viz w W Ww for

all w H and H This trivially veried

Toverify we use and to compute

W AWv W AV I v W V A v QAv

Being a partial isometry one has p WW for the pro jection p onto the image of W and

in this case W W I for the pro jection onto the subspace of H on which W is isometric this

subspace is H itself Hence follows from since



U QAU W QAW WW AWW p Ap

When Q fails to preserve the unit the ab ove construction still applies but W is no longer a

partial isometry one rather has k W k kQI k Thus it is no longer p ossible to regard H as a

subspace of H

If A and p erhaps B are nonunital the theorem holds if Q can be extended as a p ositive

map to the unitization of Asuch that the extension preserves the unit I p erhaps relative to the

unitization of B When the extension exists but do es not preserve the unit one is in the situation

of the previous paragraph

The relevance of Stinesprings theorem for quantum mechanics stems from the following result

Prop osition Let A bea commutative unital C algebra Then any positive map Q A

is completely positive

By Theorem wemay assume that A C X for some lo cally compact Hausdor space

n n

X Wemay then identify M C X with C X M C The pro of then pro ceeds in the following

steps

P

n

where F x f xM for f C X andM M C and the Elements of the form F

i i i i

i

n

sum is nite are dense in C X M C

Such F is p ositive i all f and M are p ositive

i i

n

Positive elements G of C X M C can b e normapproximated by p ositive F s ie when

G there is a sequence F suchthatlim F G

k k k

Pure states and irreducible representations

Q F ispositivewhen F is p ositive

n

Q is continuous

n

n

If F G inC X M C then QGlim QF is a normlimit of p ositive elements

k k k

hence is p ositive

Wenowproveeach of these claims

n

Take G C X M C and pick Since G is continuous the set

O fy X k Gx Gy k g

x

op en for each x X This gives an op en cover of X which by the compactness of X is

has a nite sub cover fO O g A partition of unity sub ordinate to the given cover

x x

l

is a collection of continuous p ositive functions C X where i lsuch that the

i

P

l

x for all x X Suchapartitionofunity exists and supp ort of lies in O

i i

x

i

i

n

Now dene F C X M C by

l

l

X

xGx F x

i i l

i

Since k Gx Gx k for all x O one has

i

x

i

l l l

X X X

k F x Gx kk xGx Gx k x k Gx Gx k x

l i i i i i

i i i

n

Here the norm is the matrix norm in M C Hence

k F G ksupk F x Gx k

l l

xX

n n

An element F C X M C is p ositivei F x is p ositivein M C for each x X In

n

F is p ositivei particular when F xf xM for some f C X andM M C then

n

f is p ositivein C X and M is p ositivein M C By we infer that F dened by

P

F x f xM is p ositive when all f and M are p ositive

i i i i

i

When G in item is p ositive then each Gx is p ositive as wehave just seen

i

P

On F as sp ecied in one has Q F Qf M Noweach op erator B M

n i i i

i

n

when B and M are p ositive as can be checked in a faithful repre is p ositive in M B

i

sentation Since Q is p ositive it follows that Q maps each p ositive element of the form

n

P

n

F f M into a p ositive member of M B

i i

i

Weknow from that Q is continuous the continuityof Q follows b ecause n

n

A normlimit A lim A of p ositive elements in a C algebra is p ositive b ecause by

n n

wehave A B B and lim B B exist b ecause of Finally A B B by continuity

n n n

n

of multiplication ie by

Pure states and irreducible representations

We return to the discussion at the end of One sees that the compact convex sets in the

examples have a natural b oundary The intrinsic denition of this b oundary is as follows

ELEMENTARY THEORYOF C ALGEBRAS

Denition An extreme p oint in a convex set K in some vector space is a member

of K which can only bedecomposedas



if The col lection K of extreme points in K is cal led the extreme

 e

b oundary of K An extreme point in the state space K S A of a C algebra A is cal led a

pure state A state that is not pureiscal ledamixed state

When K S A is a state spaceofaC algebra we write P A or simply P for K referred

e

to as the pure state space of A

Hence the pure states on A C C are the p ointsandin where is identied with

the functional mapping to whereas maps it to The pure states on A M C are the

matrices in for which x y z these are the pro jections onto onedimensional

subspaces of C

n

More generallywe will prove in that the state space of M C consists of all p ositive

n

matrices with unit trace the pure state space of M C then consists of all onedimensional

pro jections This precisely repro duces the notion of a pure state in quantum mechanics The

rst part of Denition is due to Minkowski it was von Neumann who recognized that this

denition is applicable to quantum mechanics

Wemaynow ask what happ ens to the GNSconstruction when the state one constructs the

representation from is pure In preparation

Denition A representation of a C algebra A on a Hilbert space H is cal led irre

ducible if a closed subspaceof H which is stable under A is either H or

This denition should b e familiar from the theory of group representations It is a deep fact of

C algebras that the qualier closed may b e omitted from this denition but we will not prove

N N

this Clearly the dening representation of the matrix algebra M on C is irreducible In

d

the innitedimensional case the dening representations of BH onH is irreducible as well

d

Prop osition Each of the fol lowing conditions is equivalent to the irreducibility of A on

H

The commutant of A in BH is fI j C g in other words A BH Schurs

lemma

Every vector in H is cyclic for A recal l that this means that A is dense in H

The commutant A is a algebra in BH so when it is nontrivial it must contain a self

adjoint element A which is not a multiple of I Using Theorem below and the sp ectral

theorem it can b e shown that the pro jections in the sp ectral resolution of A lie in A if A do es

Hence when A is nontrivial it contains a nontrivial pro jection p But then pH is stable under

A contradicting irreducibility Hence irreducible A CI

Converselywhen A CIand is reducible one nds a contradiction b ecause the pro jection

onto the alleged nontrivial stable subspace of H commutes with A Hence A CI

irreducible

When there exists a vector Hfor which A is not dense in H we can form the pro jection

onto the closure of A By Lemma with M A this pro jection lies in A so that

cannot be irreducible by Schurs lemma Hence irreducible every vector cyclic The

con verse is trivial

We are now in a p osition to answer the question p osed b efore

Theorem The GNSrepresentation A of a state SA is irreducible i is pure

When is pure yet A reducible there is a nontrivial pro jection p A by Schurs

lemma Let b e the cyclic vector for If p then Ap pA forall A A

Pure states and irreducible representations

so that p as is cyclic Similarly p is imp ossible We may then decomp ose

where and are states dened as in with p k p k

p k p k and k p k Hence cannot be pure This proves pure

irreducible

In the opp osite direction supp ose is irreducible with for SA and



Then which is p ositive hence A A A A for all A A By

 

this yields

j A B j A A B B A A B B

  

for all A B This allows us to dene a quadratic form ie a sesquilinear map Q on A by

Q A B A B



This is well dened when A A then A A A A by

  

so that

A A B j jQ A B Q A B j j

  

by in other words Q A B Q A B Similarly for B



Furthermore and imply that Q is b ounded in that

jQ j C k kk k

for all A with C It follows that Q can b e extended to all of H b ycontinuity

Moreover one has

Q Q

by with A A B and



Lemma Let a quadratic form Q on a Hilbert space H be bounded in that holds

for al l H and some constant C There is a bounded operator Q on H such that

Q Q for al l Handk Q k C When is satised Q is selfadjoint

Q is then b ounded by so that by the RieszFischer Hold xed The map

theorem there exists a unique vector suchthat Q Dene Q by Q The

selfadjointness of Q in case that holds is obvious

Now use to estimate

k Q k QQ QQ C k Q kk k

taking the supremum over all in the unit ball yields k Q k C k Q k whence k Q k C

Continuing with the pro of of we see that there is a selfadjoint op erator Q on H such

that

B A B A Q



It is the immediate from that Q C for all C A Hence Q A since is

irreducible one must have Q tI for some t R hence and show that



is prop ortional to and therefore equal to by normalization so that is pure

From wehavethe

Corollary If A H is irreducible then the GNSrepresentation A H denedby

any vector state corresponding to a unit ve ctor H is unitarily equivalent to A H

Combining this with yields

Corollary Every irreducible representation of a C algebra comes from a pure state via

the GNSconstruction

ELEMENTARY THEORYOF C ALGEBRAS

A useful reformulation of the notion of a pure state is as follows

Prop osition A state is pure i for a positive functional implies t for



some t R

We assume that A is unital if not use and For or the claim is obvious

When is pure and with thenI since is p ositive hence

k k I I I Hence Iwould imply whereas I implies

contrary to assumption Hence I and I are states and

I I

with I Since is pure bywehave I

Converselyif holds then cf the pro of of so that t by

 

assumption normalization gives t hence and is pure



The simplest application of this prop osition is

Theorem The pure state space of the commutative C algebra C X equipped with the



relative w topology is homeomorphic to X

In view of Prop osition and Theorems and we merely need to establish a

bijective corresp ondence between the pure states and the multiplicative functionals on C X



The case that X is not compact may b e reduced to the compact case by passing from A C X



to A C X cf and etc This is p ossible b ecause the unique extension of a pure

I

state on C X to a state on C X guaranteed by remains pure Moreover the extension of



amultiplicative functional dened in coincides with the extension of a state dened in

I

and the functional in clearly denes a pure state

Thus weput A C X Let C X cf the pro of of and supp ose a functional

x

satises Then ker ker and ker isanideal But ker is a maximal

x x x

ideal so when it must b e that ker ker Since two functionals on anyvector space

x

are prop ortional when they have the same kernel it follows from that is pure

x

Conversely let b e a pure state and picka g C X with g Dene a functional

X

X by f fg Since f f f g and g one on C

g g X g



has Hence t for some t R by In particular ker ker It

g g g

follows that when f ker then fg ker for all g C X since any function is a linear

combination of functions for which g Hence ker is an ideal which is maximal b ecause

X

the kernel of a functional on any vector space has co dimension Hence is multiplicative by

Theorem

It could b e that no pure states exist in S A think of an op en convex cone It would follow that

suchaC algebra has no irreducible representations Fortunately this p ossibility is excluded by

the KreinMilman theorem in functional analysis whichwe state without pro of The convex

hull coV ofasubsetV of a vector space is dened by

coV fv w j v w V g

Theorem Acompact convex set K embeddedinalocal ly convex vector space is the closure

co K of the convex hul l of its extreme points In other words K

e

It follows that arbitrary states on a C algebra may be approximated by nite convex sums

of pure states This is a sp ectacular result for example applied to C X itshows that arbitrary

probability measures on X maybeapproximated by nite convex sums of p oint Dirac measures

In general it guarantees that a C algebra has lots of pure states For example wemaynow rene

Lemma as follows

Theorem For every A A and a A there is a pure state on A for which

R a

Aa There exists a pure state such that j Aj k A k a

The C algebra of compact op erators

We extend the state in the pro of of to C A Ibymultiplicativity and continuitythatis

n n

we put A a etc It follows from that this extension is pure One easily checks that

a

the set of all extensions of to A which extensions weknow to b e states see the pro of of

a

is a closed convex subset K of S A hence it is a compact convex set By the KreinMilman

a

theorem it has at least one extreme p oint If were not an extreme p ointinS A it

a a

would b e decomp osable as in But it is clear that in that case and would coincide



A I so that cannot b e an extreme p ointofK on C

a a

Wemaynow replace the use of by in the pro of of the GelfandNeumark Theorem

concluding that the universal representation may b e replaced by We

u r

P A

may further restrict this direct sum by dening two states to b e equivalent if the corresp onding

GNSrepresentations are equivalent and taking only one pure state in each equivalence class Let

us refer to the ensuing set of pure states as P A Wethenhave

A A A

r

P A

It is obvious that the pro of of still go es through

The simplest application of this renementis

Prop osition Every nitedimensional C algebra is a direct sum of matrix algebras

F or any morphism hence certainly for any representation one has the isomorphism

A A ker Since A ker is nitedimensional it must b e that A is isomorphic to an

algebra acting on a nitedimensional vector space Furthermore it follows from Theorem

below that A A in every nitedimensional representation of A up on which

n

implies that Amust b e a matrix algebra as BH is the algebra of n n matrices for H C

Then apply the isomorphism

The C algebra of compact op erators

n

It would app ear that the appropriate generalization of the C algebra M C of n n matrices

to innitedimensional Hilb ert spaces H is the C algebra BH of all b ounded op erators on H

n

This is not the case For one thing unlike M C which as will follow from this section has

only one irreducible representation up to equivalence BH has ahuge number of inequivalent

representations even when H is separable most of these are realized on nonseparable Hilb ert

spaces

For example it follows from that any vector state on B H denes an irreducible

representation of BH which is equivalent to the dening representation On the other hand we

know from and the existence of b ounded selfadjoint op erators with continuous sp ectrum

suchasanymultiplication op erator on L X where X is connected that there are many other

pure states whose GNSrepresentation is not equivalent to the dening representation Namely

when A BH and a A but a is not in the discrete sp ectrum of A as an op erator on H

ie there is no eigenvector H for which A a then cannot b e equivalentto

a a a

a

A with eigenvalue For it is easy to show from that H is an eigenvector of

a a a

a In other words a is in the continuous sp ectrum of A Abutin the discrete sp ectrum of

A which excludes the p ossibility that A and A are equivalent as the sp ectrum is

a a

invariant under unitary transformations

Another argument against BH is that it is nonseparable in the nomtop ology even when H

n

is separable The appropriate generalization of M C to an innitedimensional Hilb ert space H

turns out to b e the C algebra B H of compact op erator on H In noncommutative geometry



elements of this C algebra play the role of innitesimals in general B H is a basic building



blo ck in the theory of C algebras This section is devoted to an exhaustive study of this C

algebra

Denition Let H be a Hilbert space The algebra B H of niterank op erators on

f

H is the nite linear span of al l nitedimensional projections on H In other words an operator

A BH lies in B H when AH fAj Hg is nitedimensional f

ELEMENTARY THEORYOF C ALGEBRAS

The C algebra B H of compact op erators on H is the normclosureofB H in BH

 f

in other words it is the smal lest C algebra of BH containing B H In particular the norm

f

in B H is the operator norm An operator A BH lies in B H when it can be

 

approximated in norm by niterank operators

It is clear that B H isa algebra since p p for any pro jection p The third item in the next

f

prop osition explains the use of the word compact in the presentcontext

Prop osition The unit operator I lies in B H i H is nitedimensional



The C algebra B H is an ideal in BH



If A B H then AB is compact in H with the normtopology Here B is the unit bal l

  

in H ie the set of al l H with k k

Firstly for any sequence or net A B H we may cho ose a unit vector A H

n f n n

Then A I so that k A I k Hence sup k A I k hence

n n n

k k

k A I k is imp ossible by denition of the norm in BH hence in B H

n 

Secondly when A B H andB BH then AB B H since AB H AH But since

f f

BA A B and B H is a algebra one has A B B H and hence BA B H

f f f

Hence B H is an ideal in BH save for the fact that it is not normclosed unless H has nite

f

dimension Nowif A A then A B AB and BA BA by continuityofmultiplication in

n n n

BH Hence B H isanidealby virtue of its denition



Thirdly note that the weak top ology on H in which i for all H

n n

is actually the w top ology under the dualityofH with itself given by the RieszFischer theorem

Hence the unit ball B is compact in the weak top ology by the BanachAlaoglu theorem So if we



can showthat A B H mapsweakly convergent sequences to normconvergent sequences then



A is continuous from H with the weak top ology to H with the normtop ology since compactness

is preserved under continuous maps it follows that AB is compact



Indeed let in the weak top ologywith k k for all n Since

n n

k k k lim k kk

n n

n

one has k k Given cho ose A B H such that k A A k and put

f f f

p A H the nitedimensional pro jection onto the image of A Then

f f

k A A kk A A A A A k k A kk p k

n f n f f n f n

 

Since the weak and the norm top ology on a nitedimensional Hilb ert space coincide they coincide

on pH sothatwe can nd N such that k p k for all nN Hence k A A k

n n

Corollary A selfadjoint operator A B H has an eigenvector with eigenvalue a

 a

such that jaj k A k

Dene f B R by f k A k When w eakly with k kthen

A  A n n

jf f j j A A A Aj k A A k j A Aj

A n A n n n n n

The rst term go es to zero by the pro of of noting that A A B H and the second go es



to zero by denition of weak convergence Hence f is continuous Since B is weakly compact

A 

f assumes its maximum at some This maximum is k A k by Now the Cauchy

A a

Schwarz inequalitywithgives k A k A A k A A k with equalityiA Ais

prop ortional to Hence when A A the prop erty k A k k A k with k k implies

a a

A a where a k A k The sp ectral theorem or the continuous functional calculus with

a a

p

f A A A implies A a Clearly jaj k A k

a a

The C algebra of compact op erators

P

Theorem A selfadjoint operator A BH is compact i A a normconvergent

i i

i

sum where each eigenvalue a has nite multiplicity Ordering the eigenvalues so that a a

i i j

when ij one has lim ja j In other words the set of eigenvalues is discrete and can

i i

only have as a possible accumulation point

This ordering is p ossible b ecause by there is a largest eigenvalue

Let A B H be selfadjoint and let p be the pro jection onto the closure of the linear



span of all eigenvectors of A As in Lemma one sees that A p so that pA pA

Hence p A I pA is selfadjoint and compact by By the compact selfadjoint

op erator p A has an eigenvector which must lie in p H and must therefore b e an eigenvector

of A in p H By assumption this eigenvector can only b e zero Hence k p A k which

implies that A restricted to p H is zero which implies that all vectors in p H are eigenvectors

with eigenvalue zero This contradicts the denition of p H unless p H This proves A

compact and selfadjoint A diagonalizable

Let A b e compact and selfadjoint hence diagonalizable Normalize the eigenvectors

i a

i

to unit length Then lim for all H since the form a basis so that

i i i

X

j j

i

i

ja j by the pro of of which clearly converges Hence weakly so k A k

i i i

Hence lim ja j This proves A compact and selfadjoint A diagonalizable

i i

with lim ja j

i i

Let now A b e selfadjoint and diagonalizable with lim ja j For N and H

i i

one then has

N

X X X X

k A a k k a k ja j j j ja j j j

i i i i i i i N i

i

iN  iN  iN 

P

N

a k b ecause lim ja j Using this is ja j so that lim k A

i i N N N N

i

P

N

Since the op erator a is clearly of nite rank this pro ves that A is compact Hence

i i

i

A selfadjoint and diagonalizable with lim ja j A compact

i i

any closed subspace of H is compact which by Finally when A is compact its restriction to

proves the claim ab out the multiplicity of the eigenvalues

We now wish to compute the state space of B H This involves the study of anumber of



subspaces of BH which are not C algebras but which are ideals of BH except for the fact

that they are not closed

Denition The Hilb ertSchmidt norm k A k of A BH is denedby

X

k Ae k k A k

i

i

H the righthand side is independent of the choice of the basis where fe g is an arbitrary basis of

i i

The Hilb ertSchmidt class B H consists of al l A BH for which k A k

The trace norm k A k of A BH is denedby





k A k k A A k





where A A is dened by the continuous functional calculus The trace class B H consists



of al l A BH for which k A k

To show that is indep endent of the basis we take a second basis fu g with corre

i i

P P

sp onding resolution of the identity I u w eakly Aince I e wethenhave

i i

i i

X X X

k Au k e u A Au e e u k A k u A Ae

i j i i j j i i j

i ij ij

ELEMENTARY THEORYOF C ALGEBRAS

If A B H then



X

Tr A e Ae

i i

i

is nite and indep endent of the basis when A B H it may happ en that Tr A dep ends on



the basis it mayeven b e nite in one basis and innite in another Converselyitcanbeshown

that A B H when Tr A where Tr is dened in terms of the decomp osition by

  

r A iTr A For A B H one has Tr A Tr A One always Tr A Tr A Tr A iT

  

 

has the equalities

k A k Tr jAj



k A k Tr jAj Tr A A

where

p

jAj A A

In particular when A one simply has k A k Tr A which do es not dep end on the basis



whether or not A B H The prop erties



Tr A A Tr AA

for all A BH and

Tr UAU Tr A

for all p ositive A BH and all unitaries U follow from by manipulations similar to

those establishing the basisindep endence of Also the linearity prop erty

TrA B Tr A T r B

for all A B B H is immediate from



It is easy to see that the Hilb ertSchmidt norm is indeed a norm and that B H is complete

in this norm The corresp onding prop erties for the trace norm are nontrivial but true and will

not b e needed In any case for all A BH onehas

k A k k A k



k A k k A k

Toprove this we use our old trick although k B kk B k for all unit vectors for every



and note that there is a H of norm suchthatk B k k B k Put B A A



k A A k k A k by Completing to a basis fe g wehave

i i

X

  

k A kk A A k A A k k A A k e k k A k

i 

i

Letting then proves Thesametrick with k A k k A k establishes

The following decomp osition will often b e used

Lemma Every operator A BH has a polar decomp osition

A U jAj

p

A A cf and U is a partial isometry with the same kernel as A where jAj

First dene U on the range of jAj by U jAjA Then compute

U jAjUjAj AA A A jAj jAj jAj

Hence U is an isometry on ranjAj In particular U is well dened for this prop erty implies that

if jAj jAj then U jAj U jAj Then extend U to the closure of ranjAjbycontinuity

 

and put U on ranjAj One easily veries that

jAj U A

and that U U is the pro jection onto the closure of ranjAj whereas UU is the pro jection onto

the closure of ranA

The C algebra of compact op erators

Prop osition One has the inclusions

B H B H B H B H BH

f  

with equalities i H is nitedimensional

P

We rst show that B H B H Let A B H Since e jAje for every

   i i

i

P

we can nd N such that e jAje Let p be the pro jection onto the linear

i i

N 

iN 

span of all e i N Using and wehave

i

k jAj p k k p jAjp kk p jAjp k



N  N  N  N  N 

p jAj in the op eratornorm top ology Since the star is normcontinuous by so that jAj

N 

this implies p jAj jAj Now p jAj obviously has nite rank for every so

N  N 

is compact by Denition Since A U jAj jAj by Prop osition that jAj

implies that A B H



The pro of that B H B H is similar this time wehave



k jAjp k k p jAj p kk p jAj p k

N  N  N  N  N 

jAj with the same conclusion so that jAjp

N 

Finallywe use Theorem to rewrite and as

X

k A k a

 i

i

X

k A a k

i

i

where the a are the eigenvalues of jAj This immediately gives

i

k A k k A k



implying B H B H



Finally the claim ab out prop er inclusions is trivially established by pro ducing examples on the

basis of and

The chain of inclusions is sometimes seen as the noncommutative analogue of



X X X X X

c 

where X is an innite discrete set Since X X and X X X this

   

analogy is strengthened by the following result

Theorem One has B H B H and B H B H BH under the pairing

   

ATr A A

is identied with A BH Here B H is identied with B H andA B H

  

The basic ingredient in the pro of is the following lemma whose pro of is based on the fact that

B H is a Hilb ert space in the inner pro duct

A B Tr A B

Toshow that this is well dened use and

Lemma For B H and A BH one has



jTr Aj k A kk k 

ELEMENTARY THEORYOF C ALGEBRAS

Using for and for the inner pro duct as well as and we

estimate

jj j jAU jj jj j jTr Aj jTr AU jj

k jj k k AU jj k k k Trjj U A AU jj



Now observethat if A A then Tr A Tr A for all A A B H since on account

  

of one has A A i all eigenvalues of A are all eigenvalues of A Then use

 

U A AU jj k AU k so from the ab ove insightwearriveat From wehave jj

U A AU jj k k k AU k k A k Trjj



since U is a partial isometry Hence wehave

Wenowprove B H B H It is clear from that

 

B H B H

 

with

k kk k



Toprovethat B H B H we use For B H and A B H B H we

   

therefore have

jAj k kk A kk kk A k

Hence B H since B H is a Hilb ert space by RieszFischer there is an op erator B H

such that A Tr A for all A B H In view of we need to sharp en B H

to B H To do so cho ose a nitedimensional pro jection p and note that pjjB H

 f

B H the presence of p even causes the sum in to b e nite in a suitable basis Nowuse



the p olar decomp osition U jj with to write

Tr pjj Tr pU Tr pU pU

changing the order inside the trace is justied by naive arguments since the sum in is

nite Using the original assumption B H wehave



jTr pjjj k kk pU kk kk p kk k

since U is a partial isometry whereas k p k in view of and p p p Now cho ose

a basis of H and take p to b e the pro jection onto the subspace spanned by the rst N elements

from and wethenhave

N

X

jTr pjjj j e jje j k k

i i

i

P

N

It follows that the sequence s j e jje j is b ounded and since it is positive it must

N i i

i

havea limit By and this means that k k k k so that B H hence

 

B H B H Combining this with and we conclude that B H B H

   

and k k k k



We turn to the pro of of B H BH It is clear from that BH B H with

 

k A kk A k

To establish the converse pick A B H and H and dene a quadratic form Q on H

 A

by

Q Aj j A

The C algebra of compact op erators

Here the op erator j j is dened by j j For example when has

unit length j j is the pro jection and in general j j k k Note that

j j j jsothat

p p

jj jj j j j j j j k kk k

Since for any pro jection pthenumber Tr p is the dimension of pH take a basis whose elements

lie either in pH or in p H wehaveTr Hence from we obtain

kj jk k kk k



Since A B H by assumption one has



jAj jj k A kk j jk



Combining and wehave

jQ j k A kk kk k

A

Hence by Lemma and there is an op erator A with

k A kk A k

such that Aj j A Now note that A Tr j jA this follows by



evaluating over a basis containing k k j Hence A j jTr j jA

Extending this equation by linearit y to the span B H of all j j and subsequently by

f

continuity to B H we obtain A Tr A Hence B H BH so that with

 

we obtain B H BH Combining and we nd k A kk A k so that the



identication of B H with BH is isometric



Corollary The state space of the C algebra B H of al l compact operators on



some Hilbert space H consists of al l density matrices whereadensity matrix is an element

B H which is positive and has unit traceTr



The pure state spaceof B H consists of al l onedimensional projections



The C algebra B H possesses only one irreducible representation up to unitary equiva



lence namely the dening one

P

Diagonalize p cf and Using A which is p ositive the condition

i i i

i

A yields p Conversely when all p the op erator is p ositive The normalization

i i

P

condition k kk k p see and yields

 i

The next item is then obvious from

Finally follows from and Corollaries and

Corollary is one of the most imp ortant results in the theory of C algebras Applied

n

to the nitedimensional case it shows that the C algebra M C of n n matrices has only one

irreducible representation

y denition The opp osite extreme to a pure state on B H isafaithful state for whichb



the leftideal N dened in is zero In other words one has Tr A A for all A

Prop osition The GNSrepresentation corresponding to a faithful state on B H is



unitarily equivalent to the representation B H on the Hilbert space B H of HilbertSchmidt



operators given by leftmultiplication ie

AB AB

ELEMENTARY THEORYOF C ALGEBRAS

It is obvious from that for A BH andB B H onehas

k AB k k A kk B k

so that the representation is welldened even for A BH rather than merely A

B H Moreover when A B B H onehas



Tr AB Tr BA

This follows from and the identity

X

n n n

AB i B i A B i A



n

 

When B H and then B H see and It is easily seen that



is cyclic for B H when is faithful Using and we compute



   

A A Tr A Tr A

The equivalence b etween and nowfollows from or



For an alternative pro of use the GNS construction itself The map A A with B H



maps B H into B H and if is faithful the closure in norm derived from the inner pro duct



of the image of this map is B H

The double commutant theorem

The socalled double commutant theorem was proved by von Neumann in and remains a

central result in op erator algebra theory For example although it is a statement ab out von

Neumann algebras it controls the irreducibility of representations Recall that the commutant

M of a collection M of b ounded op erators consists of all b ounded op erators whichcommute with

all elements of M the bicommutant M is M

We rst give the nitedimensional version of the theorem this is already nontrivial and its

pro of contains the main idea of the pro of of the innitedimensional case as well

n

Prop osition Let M bea algebra and henceaC algebra in M C containing I here

n Then M M

n

The idea of the pro of is to take n arbitrary vectors in C and given A M

 n

construct a matrix A M suchthat A A for all i n Hence A A M We

 i  i 

n

will write H for C

Cho ose some H and form the linear subspace M of H Since H is nite



dimensional this subspace is closed and we may consider the pro jection p M onto this

subspace By Lemma one has p M Hence A M commutes with p Since I Mwe

therefore haveI M so p and A AppA M Hence AA for



some A M



n n n

Nowcho ose H and regard as an elementofH HHC

 n  n

the direct sum of n copies of H where lies in the ith copy Furthermore embed M in

i

n n n

M BH by A A AI where I is the unit in M BH this is the BH

n n

n

diagonal matrix in M BH in which all diagonal entries are A

n

Now use the rst part of the pro of with the substitutions HH M M A A A

there exists A M such and Hence given and A M

  n  n

that

A A

 n   n

n n

BH compute B A B A Hence M M M It is For arbitrary B M

ij ij

n

easy to see that M M M M so that

n

M M

The double commutant theorem

Therefore A A for some A M Hence reads A A for all i n

   i  i

n

As it stands Prop osition is not valid when M C is replaced by BH where dim H

To describ e the appropriate renement we dene two top ologies on BH which are weaker

than the normtop ology wehave used so far and whose denition we rep eat for convenience

Denition The normtop ology on BH is dened by the criterion for convergence

gy is given by al l sets of the form A A i k A A k Abasis for the normtopolo

n

O AfB BH j k B A k g

where A BH and

The strong top ology on BH is denedbytheconvergence A A i k A A k

for al l H Abasis for the strong topology is given by al l sets of the form

s

O A fB BH j k B A k i ng

 n i

where A BH Hand

 n

The weak top ology on BH is denedbytheconvergence A A i j A Aj

for al l H Abasis for the weak topology is given by al l sets of the form

w

A fB BH jj A A j O i ng

 n  n i i

where A BH H and

 n  n

These top ologies should all b e seen in the light of the general theory of lo cally convex top ological

vector spaces These are vector spaces whose top ology is dened by a family fp g of seminorms

recall that a seminorm on a vector space V is a function p VR satisfying and A

net fv g in V converges to v in the top ology generated by a given i p v v for all

The normtop ology is dened by a single seminorm namely the op erator norm whichiseven

anorm Its op en sets are generated by balls in the op erator norm whereas the strong and the

weak top ologies are generated by nite intersections of balls dened by seminorms of the form

s w

p Ak A k and p AjAj resp ectively The equivalence b etween the denitions

of convergence stated in and the top ologies dened by the op en sets in question is given in

theory of lo cally convex top ological vector spaces

The estimate shows that normconvergence implies strong convergence Using the Cauch

Schwarz inequality one sees that strong convergence implies weak convergence In other

words the norm top ology is stronger than the strong top ology which in turn is stronger than the

weak top ology

Theorem Let M bea algebrainBH containing I The fol lowing areequivalent

M M

M is closed in the weak operator topology

M is closed in the strong operator topology

It is easily veried from the denition of weak convergence that the commutant N of a algebra

N is always weakly closed for if A A weakly with all A NandB Nthen

A B AB B A lim A B B A lim A B

M sothatM is weakly closed Hence If M M then M N for N

Since the weak top ology is weaker than the strong top ology is trivial

Toprove we adapt the pro of of to the innitedimensional situation Instead

of M whichmay not b e closed we consider its closure Msothatp M Hence A M

HILBERT C MODULES AND INDUCED REPRESENTATIONS

implies A M in other words for every there is an A M such that k A A k

n

For H this means that

n

X

k A A k k A A k

 n i

i

Noting the inclusion

n

X

s

f k A B k g O A

i  n

i

cf it follows that A A for Since all A M and M is strongly closed this

implies that A M so that M M With the trivial inclusion M M this proves that

M M

Hilb ert C mo dules and induced representations

Vector bundles

This chapter is concerned with the noncommutative analogue of a vector bundle Let us rst

recall the notion of an ordinary vector bundle this is a sp ecial case of the following

Denition A bundle BX F consists of topological spaces B the total space X the

base F the typical b er and a c ontinuous surjection P X with the fol lowing property



each x X has a neighbourhood N such that thereisahomeomorphism N N F

X F for which where X F X is the projection onto the rst factor

X X

F F

restricted so that The maps are called lo cal trivializations We factorize



to x provides a homeomorphism between the latter and the t ypical b er F Each subset



x is called a b er of B One may think of B as X with a copyof F attached at each p oint

The simplest example of a bundle over a base X with typical b er F is the trivial bundle

B X F with x f x According to the denition any bundle is lo cally trivial in the

sp ecied sense

Denition A vector bundle is a bund le in which

each ber is a nitedimensional vector space such that the relative topology of each ber

coincides with its topology as a vector space

 F

x F where x N is linear each local trivialization

m

A complex vector bundle is a vector bund le with typical ber C for some m N

We will generically denote vector bundles by the letter V with typical b er F V The

n n

simplest vector bundle over X with b er V C is the trivial bundle V X C This

n

bundle leads to p ossibly nontrivial subbundles as follows Recall the denition of M A in

sp ecialized to A C X in the pro of of If X is a compact Hausdor space then

n n

M C X C X M C is a C algebra Let X in addition b e connected One should verify

n

that a matrixvalued function p C X M C is an idemp otent that is p p i each pxis

n

an idemp otentinM C Such an idemp otent p denes a vector bundle V whose b er ab ove x is

p

 n

xpxC The space V inherits a top ology and a pro jection onto the rst coordinate

p

n

from X C relativetowhich all axioms for a vector bundle are satised Note that the dimension

of px is indep endentofx b ecause p is continuous and X is connected

The converse is also true

Prop osition Let V be a complex vector bund le over a connected compact Hausdor space

n m

and an idempotent p C X M C such X with typical ber C There is an integer n m

n  n

that V X C with xpxC

Vector bundles

The essence of the pro of is the construction of a complex vector bundle V suchthat V V is

n

trivial where the direct sum is dened b erwise this is the bundle X C

Following the philosophy of noncommutative geometrywenow try to describ e vector bundles

in terms of C algebras The rst step is the notion of a section of V this is a map X V

for which x x for all x X In other words a section maps a p oint in the base space into

the b er ab ove the p oint Thus one denes the space V of all continuous sections of V This is

avector space under p ointwise addition and scalar multiplication recall that eachberofV is a

vector space Moreover when X is a connected compact Hausdor space V isarightmo dule

for the commutative C algebra C X one obtains a linear action of C X onV by

R

f xf xx

R

Since C X iscommutative this is of course a leftaction as well

For example in the trivial case one has the obvious isomorphisms

m m m m

X C C X C C X C C X

m

Afancywayofsaying this is that X C isanitely generated free mo dule for C X

n

Here a free right mo dule E for an algebra A is a direct sum E A of a numb er of copies of

acts by rightmultiplication ie A itself on which A

B A A A B A B

R  n  n

If this numb er is nite one says that the free mo dule is nitely generated

When V is nontrivial one obtains V as a certain mo dication of a nitely generated free

n n

mo dule for C X For any algebra A and idemp otent p M A the action of p on A commutes

with the action by A given by rightmultiplication on each comp onent Hence the vector space

m m

p A is a right Amo dule called pro jective When m one calls p A a nitely

generated pro jective mo dule for A

n

In particular when V X C and V is the vector bundle describ ed prior to we see

p

that

n

V p C X

p

under the obvious right action of C X

This lead to the SerreSwan theorem

Theorem Let X be a connected compact Hausdor space There is a bijective corre

spondence between complex vector bund les V over X and nitely generated projective modules

E V V for C X

This is an immediate consequence of Prop osition anyvector bundle is of the form V

p

leading to V as a nitely generated pro jective C X mo dule by Conversely given sucha

p

n n

mo dule p C X one has p C X M C and therebyavector bundle V as describ ed prior

p

to

Thus wehaveachieved our goal of describing vector bundles over X purely in terms of concepts

p ertinenttotheC algebra C X Let us now add further structure

omplex vector bund le V with an inner Denition A Hermitian vector bundle is a c



product dened on each ber x which continuously depends on x More precisely for

x

al l V the function x x x lies in C X

x

Using the lo cal trivialityofV and the existence of a partition of unity it is easily shown that

any complex vector bundle over a paracompact space can be equipp ed with such a Hermitian

structure is simply given by restricting the structure Describing the bundle as V a Hermitian

p

n n

natural inner pro duct on eachberC of X C to V One may then cho ose the idemp otent

p

n n

p C X C so as to b e a pro jection with resp ect to the usual involution on C X C ie one

HILBERT C MODULES AND INDUCED REPRESENTATIONS

has p p in addition to p p Any other Hermitian structure on V may be shown to be

p

equivalent to this canonical one

There is no reason to restrict the dimension of the b ers so as to be nitedimensional A

Hilb ert bundle is dened by replacing nitedimensional vector space in by Hilb ert

space still requiring that all b ers have the same dimension whichmay b e innite A Hilb ert

bundle with nitedimensional b ers is evidently the same as a Hermitian vector bundle The

is a Hilb ert space seen as a bundle over the base space simplest example of a Hilb ert bundle

consisting of a single p oint

The following class of Hilb ert bundles will play a central role in the theory of induced group

representations

Prop osition Let H be a closed subgroup of a local ly compact group G and take a uni

tary representation U of H on a Hilbert space H Then H acts on G H by h x v



xh U hv and the quotient

H G H G H H

H

by this action is a Hilbert bund le over X GH with projection

x v x

H H

and typical ber H

Here x v is the equivalence class in G H of x v G H and x xH is the

H H H

equivalence class in GH of x G Note that the pro jection is well dened

The pro of relies on the fact that G is a bundle over GH with pro jection

xx

H

e omit implies that every q GH has a neighb our and typical b er H This fact whose pro of w

H 

ho o d N so that N N H is a dieomorphism which satises

H H

xh xh

 H

This leads to a map N N H given by x v x U xv This

H H

map is well dened b ecause of and is a lo cal trivialization of G H All required prop erties

H

are easily checked

Hilb ert C mo dules

What follows generalizes the notion of a Hilb ert bundle in suchaway that the commutative C

algebra C X is replaced by an arbitrary C algebra B This is an example of the strategy of

noncommutative geometry

Denition A Hilb ert C mo dule over a C algebra B consists of

Acomplex linear space E

A rightaction of B on E ie maps B linearly into the space of al l linear operators

R R

on E and satises AB B A for which we shal l write B B where

R R R R

E and B B

A sesquilinear map h i EE B linear in the second and antilinear in the rst entry

B

satisfying

h i h i

B

B

h B i h i B

B B

h i

B

h i

B

for al l E and B B

Hilb ert C mo dules

The space E is complete in the norm

k kkh i k

B

We say that E is a Hilb ert Bmo dule and write E
B

One checks that is indeed a norm k k equals supf h i g where the supremum

B

p

h i is a seminorm ie a norm is taken over all states on B Since each map

B

except for p ositive deniteness by the supremum is a seminorm which is actually p ositive

denite b ecause of Lemma and

The Baction on E is automatically nondegenerate the prop ertyB forallB B implies

that h i B for all B hence h i when B is unital this is follows by taking B I

B B

otherwise one uses an approximate unit in B so that by

When all conditions in are met except so that k k dened by is only a

seminorm one simply takes the quotientof E by its subspace of all null vectors and completes

obtaining a Hilb ert C mo dule in that way

It is useful to note that and imply that

hB i B h i

B B

Example Any C algebra A is a Amodule A
A over itself with hA B i A B

A

Note that the norm coincides with the C norm by

Any Hilbert space H is a Hilbert C module H
C in its inner product

Let H be a Hilbert bund le H over a compact Hausdor space X The space of continuous

sections E H of H is a Hilbert C module H
C X over B C X for

H the function h i is denedby



C X

h i x x x

x

C X



where the inner product is the one in the ber x The rightaction of C X on H is

dened by

In the third example the norm in Hisk k sup k x k where k x kx x

x

x

so that it is easily seen that E is complete

Many Hilb ert C mo dules of interest will b e constructed in the following way Recall that a

preC algebra is a algebra satisfying all prop erties of a C algebra except p erhaps completeness

GivenapreC algebra B dene a preHilb ert Bmo dule E
B as in Denition except

that the nal completeness condition is omitted

Prop osition In a preHilbert B module and hence in a Hilbert Bmodule one has the

inequalities

k B k k kk B k

k h i h i h i k

B B B

kh i k k kk k

B

Toprove one uses and For we substitute h i

B

for in the inequality h i Expanding the rst term equals h i h i h i

B B B B

Then use and replace by k k The inequality is immediate from

B can becompleted to a Hilbert Bmodule Corollary ApreHilbert B module E

One rst completes E in the norm obtaining E Using the B action on E extends

to a Baction on E The completeness of B and then allow one to extend the Bvalued

sesquilinear form on E to a Bvalued one on E It is easily checked that the required prop erties

hold bycontinuity

HILBERT C MODULES AND INDUCED REPRESENTATIONS

In Example it is almost trivial to see that A and H are the closures of A dened over

A and of a dense subspace D resp ectively

A Hilb ert C mo dule E
B denes a certain C algebra C E B whichplays an imp ortant

role in the induction theory in A map A E E for which there exists a map A E E

such that

hAi hA i

B B

for all E is called adjointable

AB AB Theorem An adjointable map is automatical ly C linear Blinear that is

for al l E and B B and bounded The adjoint of an adjointable map is unique and the

map A A denes an involution on the space C E B of al l adjointable maps on E

Equipped with this involution and with the norm dened with respect to the norm

on E thespace C E B is a C algebra

Each element A C E B satises the bound

hAAi k A k h i

B B

for al l E The dening action of C E B on E is nondegenerate We write C E B

E
B

The prop erty of C linearity is immediate To establish Blinearityone uses this also

shows that A C E B when A C E B

To prove b oundedness of a given adjointable map A x E and dene T E B by

T hA A i It is clear from that k T kk A A k so that T is b ounded On

B

the other hand since A is adjointable one has T hA Ai so that using once

B

again one has k T kk A A kk k Hence supfk T k j k kg by the principle

of uniform b oundedness here it is essential that E is complete It then follows from that

k A k

Uniqueness and involutivity of the adjointareproved as for Hilb ert spaces the former follows

from the latter in addition requires

The space C E B is normclosed as one easily veries from and that if A A

n

converges to some element which is precisely A As a normclosed space of linear maps then A

n

on a Banachspace C E B is a Banach algebra so that its satises Tocheck one

infers from and the denition of the adjointthatk A k k A A k then use Lemma

Finally it follows from and that for xed E the map A hAi

B

from C E BtoB is p ositive Replacing A by A A in and using and then

leads to

To prove the nal claim we note that for xed E the map Z hZi is in

B

C E B When the righthand side vanishes for all itmust b e that hZi for all

B

hence for Z so that Z Here we used the fact that B for all and B in the linear

span of hE Ei implies B for by it implies that h i B

B B

Under a further assumption which is by no means always met in our examples one can

E B A Hilb ert C mo dule over B is called selfdual when every completely characterize C

b ounded Blinear map EB is of the form h i for some E

B

Prop osition In a selfdual Hilbert C module E
B the C algebra C E B coincides

B

with the space LE of al l bounded C linear and Blinear maps on E

B

In view of Theorem weonlyneedto show that agiven map A LE is adjointable

Indeed for xed E dene E B by Z hAZi By selfdualitythismust

A A B

equal hZi for some whichby denition is A

B

In the context of Example one maywonder what C A A is The map A BA

given by is easily seen to map A into C A A This map is isometric hence injective

The C algebra of a Hilb ert C mo dule

Using one infers that AB AB for all A B A Hence Aisanidealin C A A

When A has a unit one therefore has C A AA A cf the pro of of

When A has no unit C A A is the socalled multiplier algebra of A One may compute

this ob ject by taking a faithful nondegenerate representation A BH it can b e shown that

C A A is isomorphic to the idealizer of AinBH this is the set of all B BH for which

BA A for all A A One thus obtains

C C X C X C X

  b

C B H B H BH

 

Eq follows by taking C X to b e the representation on L X bymultiplication op era



tors where L is dened by a measure with supp ort X and is obtained by taking B H



to b e the dening representation see the paragraph following

In Example the C algebra C H C coincides with BH b ecause every b ounded

op erator has an adjoint Its subalgebra B H of compact op erators has an analogue in the



general setting of Hilb ert C mo dules as well

The C algebra of a Hilb ert C mo dule

In preparation for the imprimitivity theorem and also as a matter of indep endent interest we

intro duce the analogue for Hilb ert C mo dules of the C algebra B H of compact op erators on



a Hilb ert space This is the C algebra most canonically asso ciated to a Hilb ert C mo dule

Denition The C algebra C E B of compactoperators on a Hilbert C module E
B



B

is the C subalgebraof C E B generated by the adjointable maps of the type T where

E and

B

Z hZi T

B

We write C E B
E
Bandcal l this a dual pair



The word compact app ears b etween quotation marks b ecause in general elements of C E B



need not be compact op erators The signicance of the notation intro duced at the end of the

denition will emerge from Theorem b elow Using the trivially proved prop erties

B B

T T

B B

T AT

A

B B

T A T

A

where A C E B one veries without diculty that C E B is a closed sided ideal in



C E B so that it is a C algebra by Theorem From and one nds the b ound

B

kk kk k k T

E B acts nondegenerately on One sees from the nal part of the pro of of Theorem that C



E

When C E B has a unit it must coincide with C E B



Prop osition When E B A see Example one has

A A A C



This leads to the dual pair A
A
A

For E H and B C see Example one obtains

C H C B H





whence the dual pair B H
H
C 

HILBERT C MODULES AND INDUCED REPRESENTATIONS

A

One has T see Since A BA is an isometric morphism the map

A A

from the linear span of all T to A dened by linear extension of T is an

isometric morphism as well It is in particular injective When A has a unit it is obvious that

is surjective in the nonunital case the existence of an approximate unit implies that the linear

span of all is dense in A Extending to C A A by continuity one sees from Corollary



that C A A A



C

Eq follows from Denition and the fact that the linear span of all T is

B H

f

A Hilb ert C mo dule E over B is called full when the collection fh i g where run

B

over E is dense in B A similar denition applies to preHilb ert C mo dules

Given a complex linear space E the conjugate space E is equal to E as a real vector space but

has the conjugate action of complex scalars

Theorem Let E be a ful l Hilbert Bmodule The expression

B

h i T

C E B

in combination with the rightaction A A where A C E B denes E as a ful l

R



Hilbert C module over C E B In other words from E
B one obtains E
C E B The

 

leftaction B B of B on E implements the isomorphism

L

C E C E B B

 

We call A C E B in the references to etc b elow one should substitute A for B when



appropriate The prop erties and follow from and Lemma

resp ectively

To prove we use with with Z and to

showthat h i implies kh i k Since h i is p ositiveby this implies

A B

B

h i hence by

B

Moreover It follows from and that each B isadjointable with resp ect to h i

L A

applying and one nds that B is a b ounded op erator on E with

L

resp ect to kk whose norm is ma jorized by the norm of B in B The map is injective b ecause

A L

E is nondegenerate as a rightBmo dule

Let E be the completion of E in k k we will shortly prove that E E It follows from

c A c

the previous paragraph that B extends to an op erator on E denoted by the same symbol

L c

and that maps B into C E A It is trivial from its denition that is a morphism Now

L c L

observe that

A

h i T

L B

for the denitions in question imply that

A B

T Z hZi T Z h i

A B

Z

E A imply that B C E A is an The fullness of E
B and the denition of C

c L c

 

isomorphism In particular it is normpreserving by Lemma

The space E is equipp ed with two norms by applying with B or with Awewrite k k

B

and kk From and one derives

A

k k k k

A B

For E wenow use the isometric nature of and to nd that

L

A

k k k k T

B

From with B A one then derives the converse inequality to so that k k k k

A B

E E asE is complete in k k by assumption The completeness of E as a Hilb ert B Hence

c B

mo dule is equivalent to the completeness of E as a Hilb ert Amo dule

Morita equivalence

Wehavenow proved Finally noticing that as a Hilb ert C mo dule over A the space E

is full by denition of C E B the pro of of Theorem is ready



For later reference we record the remarkable identity

h i Z hZi

B

C E B

which is a restatement of

Morita equivalence

The imprimitivity theorem establishes an isomorphism b etween the resp ective representation the

ories of two C algebras that stand in a certain equivalence relation to each other

Denition Two C algebras A and B are Moritaequivalent when there exists a ful l

M

Hilbert C module E
B under which A C E B We write A B and A
E
B



Prop osition Morita equivalenceisanequivalencerelation in the class of al l C algebras

M

The reexivity prop erty B B follows from which establishes the dual pair B
B

B Symmetry is implied by proving that A
E
B implies B
E
A

M M

The pro of of transitivityismoreinvolved When A B and B C wehavethechain of dual

pairs

A
E
B
E
C



We then form the linear space E E which is the quotientofE E by the ideal I generated

 B  B

byallvectors of the form B B which carries a rightaction Cgiven by

 

R

C C

 B  B

R

on E E by Moreover we can dene a sesquilinear map h i

 B

C

h i h h i i

 B  B   B C

C

With this satises and as explained prior to one may therefore construct

a Hilb ert C mo dule denoted by E
C Remarkably if one lo oks at as dened on

E E thenull space of is easily seen to contain I but in fact coincides with it so that

 B

in constructing E one only needs to complete E E

 B

A the op erator Apart from the rightaction C the space E carries a leftaction

R

L

A A

 B  B

L

is b ounded on E E and extends to E Wenow claim that

 B

A E C C



L

Using the denition of and it is easily shown that

B

B

h h i i T

 B   B B  B

L

h i

B

Now use the assumption C E C B as in with B C and E E this yields



C

h i T Substituting this in the righthand side of and using with B C

B

the righthand side of b ecomes h h i i Using B B see

 B   B C L

with B C and with B Cweeventually obtain

C B

T T

L

h i

B B

B

This leads to the inclusion C E C A To prove the opp osite inclusion one picks a



L

P

N

C i i

E C One is an approximate unit in B C g such that T double sequence f

i i



i

HILBERT C MODULES AND INDUCED REPRESENTATIONS

P

N

i i

has lim h Zi Z from and a short computation using with then

N C

i

yields

N

X

C B

T T lim

i i

L

B B

N

i

Hence A C E C and combining b oth inclusions one nds



L

M

Therefore one has the dual pair A
E
C implying that A C This prov es transitiv

ity

Here is a simple example of this concept

Prop osition The C algebra B H of compact operators is Moritaequivalent to C with



n

dual pair B H
H
C In particular the matrix algebra M C is Moritaequivalent to C



n n

This is immediate from In the nitedimensional case one has M C
C
C where

n n

M C and C act on C in the usual way The double Hilb ert C mo dule structure is completed

by sp ecifying

i i

hz wi z w

C

i j

n

hz wi z w

ij

M C

from which one easily veries

M M M

n m n m

Since M C C and C M C one has M C M C This equivalence is imple

n nm m nm

mented by the dual pair M C
M C
M C where M C is the space of complex

matrices with n rows and m columns Weleave the details as an exercise

In practice the following way to construct dual pairs and therefore Morita equivalences is

useful

Prop osition Suppose one has

two preC algebras A and B

a ful l preHilbert Bmodule E

a leftaction of A on E such that E can be made into a ful l preHilbert Amodule with respect

to the rightaction A A

R

the identity

h i Z hZi

 

A B

for al l Z E relating the two Hilbert C module structures

the bounds

hB B i k B k h i

 

A A

k A k h i hAAi

 

B B

for al l A A and B B

M

Then A B with dual pair A
E
B where E is the completion of E as a Hilbert Bmodule

Using Corollary we rst complete E to a Hilb ert Bmo dule E By which implies

k A kk A kk k for all A A and E the action of A on E extends to an action of A on

E to a Hilb ert Amo dule E Similarlywe complete E by the leftaction B B

c L

extends to an action of B on E As in the pro of of Theorem one derives and its

c

E of converse for E so that the Bcompletion E of E coincides with the Acompletion E that

c

E E is c

Rieel induction

Since E is a full preHilb ert Amo dule the Aaction on E is injective hence faithful It follows

from Theorem and once again the fullness of E that A C E B In particular



each A A automatically satises

Clearly is inspired by into whichit is turned after use of this prop osition We

will rep eatedly use Prop osition in what follows see and

Rieel induction

Toformulate and prove the imprimitivity theorem we need a basic technique whichisofinterest

also in a more general context Given a Hilb ert Bmo dule E the goal of the Rieel induction

pro cedure describ ed in this section is to construct a representation of C E B from a repre

sentation of B In order to explicate that the induction pro cedure is a generalization of the

GNSconstruction we rst induce from a state on B rather than from a representation

Construction Suppose one has a Hilbert C mo dule E
B

g

Given a state on B dene the sesquilinear form on E by



h i

B



Since and h i arepositive cf this form is positive semidenite Itsnullspace

B

is

N f Ej g



g g

The form projects to an inner product on the quotient E N If V E EN



is the canonical projection then by denition

V V



The Hilbert space H is the closureof E N in this inner product

The representation C E B is rstly denedonE N H by

AV V A

it fol lows that is continuous Since E N is dense in H the operator A may be

denedonallofH by continuous extension of where it satises

The GNSconstruction is a sp ecial case of obtained by cho osing E B A as

explained in Example

The analogue of of course applies here The continuityof follows from and

which imply that k AV k AA Using and in succes



sion one nds that

k A kk A k

On the other hand from the pro of of Theorem one sees that

k A k supfj A Aj j SAg

Applying to B used with the denition of k A k for A C E B implies that

k A ksupfk A k SBg

A similar argumentcombined with Corollary shows that is faithful hence normpreserving

when is As a corollary one infers a useful prop ertywhich will b e used eg in the pro of of

Theorem

HILBERT C MODULES AND INDUCED REPRESENTATIONS

Lemma Let A C E B satisfy hAi for al l E Then A

B

Take a faithful state on B the condition implies that A

When one starts from a representation B rather than from a state the general construction

lo oks as follows

Construction Start from a Hilbert C module E
B

Givenarepresentation B on a Hilbert space H with inner product the sesquilinear

form is denedonEH algebraic tensor product by sesquilinear extension of



v w v h i w

B



where v w H This form is positive semidenite because and h i are The nul l

B

spaceis

N f E H j g



As in we may equal ly wel l write

E H g N f E H j



The form projects to an inner product on the quotient EH N denedby



V V



where V EH EH N is the canonical projection The Hilbert space H is the

closureof EH N in this inner product

The representation C E B is then denedonH by continuous extension of

V A I AV

where I is the unit operator on H the extension in question is possible since

k A kk A k

To prove that the form dened in is p ositive semidenite we assume that B is cyclic

P

if not the argumentbelowisrepeatedforeach cyclic summand see With v

i i

i

B one then uses and and v B where is a cyclic vector for

i i

P

to nd v h i v with B Hence by and the

B i i

 

i

p ositivityof B BH

k v hAAi v from and ac Similarly one computes k AV

B

cording to and the prop erty k A kk A k cf the text after this is b ounded by

k A k v h i v Since the second factor equals k V k this proves

B

Paraphrasing the comment after the rst version of the construction is faithful when is

Also it is not dicult to verify that is nondegenerate when is

Tointerrelate the ab ovetwo formulations one assumes that is cyclic with cyclic vector

Then dene a linear map U EEH by

U

According to and this map has the prop erty

U U





By and the map U therefore quotients to a unitary isomorphism U H H which

by and duly intertwines and

Of course any subspace of C E Bmay b e sub jected to the induced representation This

particularly applies when one has a given pre C algebra A and a homomorphism A

C E B leading to the representation AonH Further to an earlier comment one veries

that is nondegenerate when and are With slight abuse of notation w e will write A

for A The situation is depicted in Figure

The imprimitivity theorem

A E B H

H

H

H

H

induction

H

H

H

H

Hj

H

Figure Rieel induction

A E B H

R

E A H H B

R

B H

Figure Quantum imprimitivity theorem H H and

The imprimitivity theorem

After this preparation wepasstotheimprimitivitytheorem

Theorem There is a bijective correspondence between the nondegenerate representations

of Moritaequivalent C algebras A and B preserving direct sums and irreducibility This corre

spondence is as fol lows

Let the pertinent dual pair be A
E
B When A isarepresentation on a Hilbert space

H there exists a representation B on a Hilbert space H such that is equivalent to the

Rieelinducedrepresentation dened by and the above dual pair

In the opposite direction a given representation B is equivalent to the Rieelinduced

representation denedwithrespect to some representation A and the dual pair B
E
A

Taking A A as just dened one has B B Conversely taking B

B one has A A

See Figure Starting with B we construct AwithRieel induction from the dual

pair A
E
B relab el this representation as A and move on to construct B from

Rieel induction with resp ect to the dual pair B
E
A We then construct a unitary map

U H H whichintertwines and

EEH H by linear extension of We rst dene U

U v h i v

B

HILBERT C MODULES AND INDUCED REPRESENTATIONS

Note that U is indeed C linear Using the prop erties and with

and one obtains

B

U v U v v T v

    



Now use the assumption A C E B to use and subsequently and all



thand side of is then seen to be equal to V read from right to left The righ



v h i V v Now put and H H and use and

  A

from right to left with This shows that

U v U v V V v V V v

     

In particular U annihilates where EH whenever N or V N Hence

we see from the construction rstly of H H from EH and secondly of H from EH

cf that U descends to an isometry U H H dened by linear extension of

UV V v U v h i v

B

Using the assumptions that the Hilb ert C mo dule E
B is full and that the representation

B is nondegenerate we see that the range of U and hence of U is dense in H sothatU is

unitary

Toverify that U intertwines and we use and with to compute

U B V V v h B i v

L B

E is as dened in Thus writing B B where the leftaction of B B on

L

using with and from right to left the righthand side of is seen

to b e B UV V v Hence U B B U for all B B

Using the pro of that the Morita equivalence relation is symmetric see one immediately

sees that the construction works in the opp osite direction as well

It is easy to verify that leads to This also proves that the

bijective corresp ondence B A preserves irreducibility when is irreducible and

isnt one puts as ab ove decomp oses then decomp oses the induced

representation Bas and thus arrives at a contradiction since

Combined with Prop osition this theorem leads to a new pro of of Corollary

Group C algebras

In many interesting applications and also in the theory of induced representation as originally

the C algebra B featuring in the denition of a formulated for groups byFrob enius and Mackey

Hilb ert C mo dule and in Rieel induction is a socalled group C algebra

We start with the denition of the group algebra C G of a nite group with nG elements

one then usually writes C G instead of C G As a vector space C G consist of all complex

nG

valued functions on G so that C G C This is made into a algebra by the convolution

X

f g x f y g z

yzGjyzx

and the involution



f x f x

It is easy to check that the multiplication is asso ciative as a consequence of the asso ciativityof

the pro duct in G In similar vein the op eration dened byisaninvolution b ecause of the

    

prop erties x x and xy y x at the group level

A representation of C G on a Hilb ert space H is dened as a morphism C G BH

Group C algebras

Prop osition There is a bijective correspondencebetween nondegenerate representations

of the algebra C G and unitary representations U of G which preserves unitary equivalence and

direct sums and thereforepreserves irreducibility This correspondence is given in one direction

by

X

f f xU x

xG

and in the other by

U x

x

where y xy

x

It is elementary to verify that is indeed a representation of C G when U is a unitary

representation of Gandvice versa Putting x e in yields I so that cannot b e

e

degenerate

When U xVU xV for all x G then evidently f V f V for all f C G

 

f f for all f i U x The converse follows by cho osing f Similarly f

 x

U x U x for all x



WecandeneaC norm on C Gby taking any faithful representation and putting k f kk

f k Since C G is a nitedimensional vector space it is complete in this norm which therefore

is indep endentofthechoice of by Corollary

Let now G b e an arbitrary lo cally compact group such as a nitedimensional Lie group We

also assume that G is unimo dularthat is each left Haar measure is also rightinvariant This

assumption is not necessary but simplies most of the formulae We denote Haar measure by dx

it is unique up to normalization Unimo dularity implies that the Haar measure is invariantunder

R



inversion x x When G is compact we cho ose the normalization so that dx The

G



Banach space L G and the Hilb ert space L G are dened with resp ect to the Haar measure

The convolution pro duct is dened initially on C G by

c

Z



f g x dy f xy g y

G

it is evident that for a nite group this expression sp ecializes to The involution is given by

As in the nite case one checks that these op erations make C Ga algebra this time

c

one needs the invariance of the Haar measure at various steps of the pro of



ations and arecontinuous in the L norm one has Prop osition The oper

k f g k k f k k g k

  

k f k k f k

 



Hence L G is a Banach algebra under the continuous extensions of and from



C G to L G

c

Recall the denition of a Banach algebra b elow



It is obvious from invariance of the Haar measure under x x that k f k k f k so that

 

the involution is certainly continuous The pro of of is a straightforward generalization of

the case G R cf This time wehave

Z Z Z Z

 

k f g k dx j dy f xy g y j dy jg y j dx jf xy j



G G G G

Z Z

dy jg y j dx jf xj k f k k g k

 

G G

which is



C norm we construct a faithful representation on a Hilb ert In order to equip L Gwitha space

HILBERT C MODULES AND INDUCED REPRESENTATIONS



Prop osition For f L G the operator f on L G denedby

L

f f

L



is bounded satisfying k f kk f k The linear map L G BL G is a faithful

L  L



representation of L Gseen as a Banach algebra as in

Intro ducing the leftregular representation U of G on L Gby

L



U y xy x

L

it follows that

Z

f dx f xU x

L L

G

The b oundedness of f then follows from Lemma b elow One easily veries that f g

L L

f g and f f

L L L L



To provethat is faithful werstshowthatL G p ossesses the analogue of an approximate

L

see for C algebras When G is nite the deltafunction is a unit in C G For unit

e

general lo cally compact groups one would like to take the Dirac function as a unit but this



distribution is not in L G



Lemma The Banach algebra L G has an approximate unit I in the sense that



hold for al l A L Gandk kk k





Pick a basis of neighbourhoods N of e so that each N is invariantunder x x this basis is

whichisthecharacteristic function of N times partially ordered by inclusion Take I N

N

a normalization factor ensuring that k I k Eq then holds by virtue of and the



invariance of N under inversion By construction the inequality holds as an equality One

R R

 

dy f xy For f C G one therefore dy f y x and f I xN has I f xN

c

N N

has lim I f f and lim f I f pointwise ie for xed x The Leb esgue dominated



convergence theorem then leads to for all A C G and therefore for all A L G since

c



C G is dense in L G

c

To nish the pro of of wenow note from that f implies f for all

L

L G and hence certainly for I Hence k f k by Lemma so that f and



is injective

L

G Denition The reduced group C algebra C G is the smal lest C algebrainBL

r

containing C G In other words C G is the closure of the latter in the norm

L c

r

k f k k f k

r L

Perhaps the simplest example of a reduced group algebra is obtained by taking G R

Prop osition One has the isomorphism

R C R C



r



It follows from the discussion preceding that the Fourier transform maps L G

into a subspace of C R which separates p oints on R It is clear that for every p R there is an





f L R for which f p In order to apply Lemma we need to verify that

k f k k f k

r

Since the Fourier transform turns convolution into p ointwise multiplication the leftregular repre

sentation on L R isFouriertransformed into the action on L Rbymultiplication op erators

L

Hence follows from Lemma

Group C algebras

This example generalizes to arbitrary lo cally compact ab elian groups Let G b e the set of all

irreducible unitary representations U of Gsuch representations are necessarily onedimensional

so that G is nothing but the set of characters on G The generalized Fourier transform f of



f L G is a function on G dened as

Z

f dx f xU x

G

By the same arguments as for G R one obtains

C G C G



r

w found a C We return to the general case where G is not necessarily ab elian Wehaveno

algebra whichmay play the role of C G for lo cally compact groups Unfortunately the analogue

of Prop osition only holds for a limited class of groups Hence we need a dierent construction

Let us agree that here and in what follows a unitary representation of a top ological group is always

meanttobecontinuous

Lemma Let U be an arbitrary unitary representation of G on a Hilbert space H Then

f denedby

Z

f dx f xU x

G

is bounded with

k f kk f k



The integral is most simply dened weakly that is by its matrix elements

Z

f dx f xUx

G

p

Since U is unitarywehave jf j F F for all H where F xk k jf xj

L G

The CauchySchwarz inequality then leads to jf j k f k k k Lemma then



leads to

Alternatively one may dene as a Bo chner integral We explain this notion in a more

general context

Denition Let X beameasurespace and let B be a Banach space A function f X B

is Bochnerintegrable with respect to a measure on X i

the function x f x is f is weakly measurable that is for each functional B

measurable

thereisanullsetX X such that ff xjx X nX g is separable

 

the function denedbyx k f x k is integrable

It will always b e directly clear from this whether a given op erator or vectorvalued integral

maybereadasa Bochner integral if not it is understo o d as a weak integral in a sense always

R

obvious from the context The Bo chner integral dxf x can b e manipulated as if it were an

X

ordinary Leb esgue integral For example one has

Z Z

k d x f x k dx k f x k

X X

Thus reading as a Bo chner integral is immediate from

The following result generalizes the corresp ondence b etween U in and in to

L L

arbitrary representations

HILBERT C MODULES AND INDUCED REPRESENTATIONS

Theorem There is a bijective correspondencebetween nondegenerate representations of



the Banach algebra L G which satisfy and unitary representations U of G This corre

spondence is given in one direction by and in the other by

x

U x f f

x 

where f y f x y This bijection preserves direct sums and therefore irreducibility

Recall from that any nondegenerate representation of a C algebra is a direct sum of



cyclic representations the pro of also applies to L G Thus in stands for a cyclic vector

of a certain cyclic summand of H and denes U on a dense subspace of this summand it

will b e shown that U is unitary so that it can b e extended to all of H bycontinuity

Given U it follows from easy calculations that f in indeed denes a representation

It is b ounded by Lemma The pro of of nondegeneracy makes use of Lemma Since

is continuous one has lim I I stronglyproving that must b e nondegenerate

To go in the opp osite direction we use the approximate unit once more it follows from

x

from which the continuityofU is obvious that U x f lim I f Hence U x

x

lim I strongly on a dense domain The prop erty U xU y U xy thenfollows from

and The unitarityofeach U xfollows by direct calculation or from the following argument

x x 

Since k I kk I k we infer that k U x k forallx Hence also k U x k which



 

is the same as k U x k We see that U x and U x are b oth contractions this is only

p ossible when U x is unitary

FinallyifU is reducible there is a pro jection E such that E U x for all x G It follows

from that f E for all f hence is reducible Converselyif is reducible then

x

E I for all x G by the previous paragraph this implies E U x for all x The

nal claim then follows from Schurs lemma

This theorem suggests lo oking at a dierent ob ject from C G Inspired by one puts

r

Denition The group C algebra C G is the closureoftheBanach algebra algebra



L G in the norm

k f kk f k

u



where is the direct sum of al l nondegenerate representations of L G which areboundedas

u

in



Equivalently C G is the closureofL G in the norm

k f k sup k f k



where the sum is over al l representations L G of the form in which U is an irre

ducible unitary representation of G and only one representative of each equivalenceclassof such

representations is included

The equivalence b etween the two denitions follows from and Theorem

Theorem There is a bijective correspondence between nondegenerate representations

of the C algebra C G and unitary representations U of G given by continuous extension of

and This correspondencepreserves irreducibility



It is obvious from and that for any representation C G and f L G one

has

k f kk f kk f k





Hence the restriction L G satises and therefore corresp onds to U G by Theorem



Converselygiven U G one nds L G satisfying by it then follows from

that one may extend to a representation of C Gbycontinuity

C dynamical systems and crossed pro ducts

In conjunction with the second denition of C G stated in implies that for

ab elian groups C Galways coincides with C G The reason is that for G one has f

r

f C so that the norms and coincide In particular one has

n n

C R C R



For general lo cally compact groups lo oking at we see that

C G C G C G ker

L L

r

A Lie group group is said to b e amenable when the equality C GC G holds in other

r

words C G is faithful i G is amenable Wehave just seen that all lo cally compact ab elian

L

groups are amenable It follows from the PeterWeyl theorem that all compact groups are amenable

as well However noncompact semisimple Lie groups are not amenable

C dynamical systems and crossed pro ducts

An automorphism of a C algebra A is an isomorphism between A and A It follows from

Denitions and that A A for any A A any automorphism hence

k A kk A k

by

One A has a unit one has

I I

by and the uniqueness of the unit When A has no unit one may extend to an automor

I

phism of the unitization A by

I

I

A I AI

Denition An automorphic action of a group G on a C algebra A is a group homo

morphism x such that each is an automorphism of A In other words one has

x x

A A

x y xy

AB A A

x x x

A A

for al l x y G and A B A

A C dynamical system G A consists of a local ly compact group GaC algebra A and

an automorphic action of G on A such that for each A A the function from G to A denedby

x k A kiscontinuous

x

The term dynamical system comes from the example G R and A C S where R acts on S



t and f t Hence a general C dynamical system is a noncommutative by t

t

analogue of a dynamical system



Prop osition Let G A be a C dynamical system and dene L G A as the space

of al l measurable functions f G A for which

Z

k f k dx k f x k



G

is nite The operations

Z



f g x dy f y g y x

y

G



f x f x

x



turn L G A into a Banach algebra

HILBERT C MODULES AND INDUCED REPRESENTATIONS

As usual wehave assumed that G is unimo dular with a slight mo dication one may extend

these formulae to the nonunimo dular case The integral is dened as a Bo chner integral

the assumptions in Denition are satised as a consequence of the continuity assumption in

the denition of a C dynamical system Toverify the prop erties and one follows the

 

same derivation as for L G using and The completeness of L G A is proved

 

as in the case A C for which L G AL G

In order to generalize Theorem we need

Denition A covariant representation of a C dynamical system G A consists of

apair U where U is a unitary representation of G and is a nondegenerate representation

of A which for al l x G and A A satises

U x AU x A

x

Here is an elegant and useful metho d to construct covariant representations

Prop osition Let G A beaC dynamical system and suppose one has a state on A

which is Ginvariant in the sense that

A A

x

for al l x G and A A Consider the GNSrepresentation A on a Hilbert space H with

cyclic vector For x G dene an operator U x on the dense subspace A of H by

U x A A

x

This operator is wel l dened and denes a unitary representation of G on H

A B A B by Hence A B A If A B then

x x

B by so that k A B k by Hence A

x x

B so that U x A U x B

x

Furthermore implies that U xU y U xy whereas and imply that

U x A Ux B A B

This shows rstly that U x is b ounded on A so that it may be extended to H by

continuity Secondly U x is a partial isometry which is unitary from H to the closure of

U xH Taking A B in one sees that U xH A whose closure is H

x

b ecause is cyclic Hence U x is unitary

Note that with or implies that

U x

Prop osition describ es the way unitary representations of the Poincare group are con

structed in algebraic quantum eld theory in which is then taken to be the vacuum state on

the algebra of lo cal observables of the system in question Note however that not all covariant

representations of a C dynamical system arise in this way a given unitary representation U G

maymay not contain the trivial representation as a subrepresentation cf

In any case the generalization of Theorem is as follows Recall

Theorem Let G A be a C dynamical system There is a bijective correspondence



between nondegenerate representations of the Banach algebra L G A which satisfy

and covariant representations U G A This correspondence is given in one direction by

Z

dx f xU x f G

Transformation group C algebras



in the other direction one denes Af x Af x and f y f x y and puts

x x

U x f f

x

A f Af

where is a cyclic vector for a cyclic summand of C G A

This bijection preserves direct sums and thereforeirreducibility



The pro of of this theorem is analogous to that of The approximate unit in L G A



is constructed by taking the tensor pro duct of an approximate unit in L G and an approximate

unit in A The rest of the pro of may then essentially b e read o from

Generalizing weput

Denition Let G A bea C dynamical system The crossed pro duct C G A of



G and A is the closure of the Banach algebra algebra L G A in the norm

k f kk f k

u



where is the direct sum of al l nondegenerate representations of L G A which arebounded

u

as in



Equivalently C G A is the closureof L G A in the norm

k f k sup k f k



where the sum is over al l representations L G A of the form in which U is an

irreducible covariant representation of G A and only one representative of each equivalence

class of such representations is included

Here wesimplysay that a covariant representation U is irreducible when the only b ounded

A is amultiple of the unit The equivalence b etween op erator commuting with all U x and

the two denitions follows from and Theorem

Theorem Let G A be a C dynamical system There is a bijective correspondence

between nondegenerate representations of the crossedproduct C G A and covariant repre

sentations U G A This correspondence is given by continuous extension of and

This corresp ondencepreserves direct sums and thereforeirreducibility

The pro of is identical to that of

Transformation group C algebras

Wenow come to an imp ortant class of crossed pro ducts in which A C Q where Q is a lo cally



compact Hausdor space and is dened as follows

x

Denition A left action L of a group G on a space Q is a map L G Q Q satisfying

Le q q and Lx Ly q Lxy q for al l q Q and x y G If G and Q arelocal ly compact

G is a Lie group and Q is a manifold we assume that L is we assume that L is continuous If

smooth We write L q xq Lx q

x

We assume the reader is familiar with this concept at least at a heuristic level The main

example we shall consider is the canonical action of G on the coset space GH where H is a

closed subgroup of G This action is given by

xy xy

H H

SO is the subgroup where x xH cf etc For example when G SO and H

H

of rotations around the z axis one may identify GH with the unit twosphere S in R The

SOaction is then simply the usual action on R restricted to S

HILBERT C MODULES AND INDUCED REPRESENTATIONS

Assume that Q is a lo cally compact Hausdor space so that one may form the commutative

C algebra C Q cf A Gaction on Q leads to an automorphic action of G on C Q given

 

by



f q f x q

x

Using the fact that G is lo cally compact so that e has a basis of compact neighb ourho o ds it is

easy to prove that the continuityofthe Gaction on Q implies that

lim k f f k

x

xe

for all f C Q Since C Q is dense in C Q in the supnorm the same is true for f C Q

c c  

Hence the function x f from G to C Qiscontinuous at e as f f Using and

x  e

one sees that this function is continuous on all of G Hence G C QisaC dynamical



system

It is quite instructivetolookatcovariant representations U ofG C Q in the sp ecial



case that G is a Lie group and Q is a manifold Firstlygiven a unitary representation U of a Lie

group G on a Hilb ert space H one can construct a representation of the Lie algebra g by

d

dU U ExptX X

jt

dt

When H is innitedimensional this denes an unb ounded op erator which is not dened on all

of H Eq makes sense when is a smo oth vector for a U this is an element H

for which the map x U x from G to H is smo oth It can be shown that the set H of

U

smo oth vectors for U is a dense linear subspace of H and that the op erator idU X is essentially

selfadjointonH Moreover on H one has

U U

dU X dUY dU X Y

Secondly given a Lie group action one denes a linear map X from g to the space of all

X

vector elds on Q by

d

f q f ExptX q

X

jt

dt

where Exp g G is the usual exp onential map

The meaning of the covariance condition on the pair U may now be claried by

reexpressing it in innitesimal form For X g f C Q and Rnfg weput

c

Q X idU X

Q f f

From the commutativit yofC Q and resp ectivelywe then obtain



i

Q f Q g

i

Q X Q Y Q X Y

i

f Q X Q f Q

X

and may be seen as a generalization of the canonical These equations hold on the domain H

U

n

commutation relations of quantum mechanics To see this consider the case G Q R

n

where the Gaction is given by Lx q q x If X T is the k th generator of R one has

k

k l n

q Taking f q the l th coordinate function on R one therefore obtains

k T

k

l l

q The relations then b ecome

k

k

i

k l

Q q Q q

i

T T Q Q

l k

i

l l

q T Q Q

k

k

Transformation group C algebras

k

Hence one may identify Q q andQ T with the quantum p osition and momentum observables

k

k

resp ectively It should b e remarked that Q q is an unb ounded op erator but one mayshow

k

from the representation theory of the Heisenb erg group that Q q andQ T always p ossess a

k

common dense domain on which are valid

Denition Let L be a continuous action of a local ly compact group on a local ly compact

space Q The transformation group C algebra C G Q is the crossedproduct C G C Q



dened by the automorphic action

Conventionallythe Gaction L on Q is not indicated in the notation C G Q although the

construction clearly dep ends on it



One may identify L G C Q with a subspace of the space of all measurable functions from





G Q to C an element f of the latter denes F L G C Q by F x f x Clearly





L G C Q is then identied with the space of all such functions f for which



Z

k f k dx sup jf x q j



q Q

G

is nite cf In this realization the op erations and read

Z

 

f g x q dy f y qg y x y q

G

 

f x q f x x q

As always G is here assumed to b e unimo dular Here is a simple example

Prop osition Let a local ly compact group G act on Q G by Lx y xy Then

C G G B L G as C algebras



We start from C G G regarded as a dense subalgebra of C G G We dene a linear map

c

C G G BL G by

c

Z



dy f xy xy f x

G

One veries from and that f g f g and f f so that is

a representation of the algebra C G G It is easily veried that the Hilb ertSchmidtnorm

c

of f is

Z Z



k f k dx dy jf xy xj

G G

C G G we conclude from that C G G Since this is clearly nite for f

c c

B L G Since C G G is dense in B L G in the Hilb ertSchmidtnorm whichisa

 c

standard fact of Hilb ert space theory and B L G is dense in B L G in the usual op erator



norm since by Denition even B L G is dense in B L G we conclude that the

f 

closure of C G G in the op erator norm coincides with B L G

c 

Since is evidently faithful the equality C G G B L G and therefore the iso



morphism C G G B L G follows from the previous paragraph if we can show that the



norm dened by coincides with the op erator norm of This in turn is the case if all

irreducible representations of the algebra C G G are unitarily equivalentto

c

Toprove this we pro ceed as in Prop osition in whichwetake A C G G B C

c

and E C G The preHilb ert C mo dule C G
C is dened by the obvious C action on

c c

C G and the inner pro duct

c

h i

C

L G

The leftaction of A on E is as dened in whereas the C G Gvalued inner pro duct

c

on C G is given by

c



h i y x y

C GG

c

It is not necessary to consider the b ounds and Following the pro of of Theorem

one shows directly that there is a bijective corresp ondence b etween the representations of

C G GandofC c

HILBERT C MODULES AND INDUCED REPRESENTATIONS

The abstract transitive imprimitivity theorem

We sp ecialize to the case where Q GH whereH is a closed subgroup of G and the Gaction

on GH is given by This leads to the transformation group C algebra C G GH

Theorem The transformation group C algebra C G GH is Moritaequivalent to C H

We need to construct a full Hilb ert C mo dule E
C H forwhich C E C H is isomor



phic to C G GH This will b e done on the basis of Prop osition For simplicitywe assume

that b oth G and H are unimo dular In we take

A C G GH seen as a dense subalgebra of A C G GH as explained prior to

c

B C H seen as a dense subalgebra of B C H

c

E C G

c

Wemake a preHilb ert C H mo dule C G
C H by means of the rightaction

c c c

Z



f f x dh xh f h

R

H

Here f C H and C G The C H valued inner pro duct on C G is dened by

c c c c

Z

h i h dx xxh

C H

c

G

Interestingly b oth formulae may b e written in terms of the rightregular representation U of H

R

on L G given by

U hxxh

R

Namelyonehas

Z



f dh f hU h

R

H

which should b e compared with and

h i h Uh

C H L G

c

The prop erties and are easily veried from and resp ectively To prove

wetakeavector state on C H with corresp onding unit vector H Hence for



f C H L H onehas

c

Z

f f dh f h U h

H

where U is the unitary representation of H corresp onding to C H see Theorem

with G H We note that the Haar measure on G and the one on H dene a unique measure

on GH satisfying

Z Z Z

d q dx f x dh f sq h

GH G H

for any f C G and any measurable map s GH G for which s id where G GH

c

is the canonical pro jection xx xH Combining and we nd

H

Z Z

h i d q k dh sq hU h k

C H

c

GH H

es that h i is p ositive for all representations of Since this is p ositive this prov

C H

c

C H so that h i is p ositivein C H by Corollary This proves Condition

C H c

Induced group representations

easily follows from as well since h i implies that the righthand side

C H

c

of vanishes for all This implies that the function q h sq h vanishes almost

everywhere for arbitrary sections s Since one maycho ose s so as to b e piecewise continuous and

C G this implies that

c

Wenow come to the leftaction of A C G GH onC G and the C G GH valued

L c c c

C G These are given by inner pro duct h i on

c

C GGH

c

Z



f x dy f xy x y

L H

G

Z



h i x y dh yh x yh

H

C GGH

c

H

Using and one may check that is indeed a leftaction and that C G

L c

C G GH is a preHilb ert C mo dule with resp ect to the rightaction of C G GH given by

c c

f f cf Also using and it is easy to verify

R L

the crucial condition

To complete the pro of one needs to show that the Hilb ert C mo dules C G
C H and

c c

C G
C G GH are full and that the b ounds and are satised This is indeed

c c

the case but an argument that is suciently elementary for inclusion in these notes do es not

seem to exist Enthusiastic readers may nd the pro of in MA Rieel Induced representations of

C algebras Adv Math

Induced group representations

The theory of induced group representations provides a mechanism for constructing a unitary

representation of a lo cally compact group G from a unitary representation of some closed subgroup

H Theorem then turns out to be equivalent to a complete characterization of induced

group representations in the sense that it gives a necessary and sucient criterion for a unitary

representation to b e induced

In order to explain the idea of an induced group representation from a geometric p oint of view

we return to Prop osition The group G acts on the Hilb ert bundle H dened by by

means of

U xy v xy v

H H

Since the leftaction x y xy of G on itself commutes with the rightaction h y yh of H on

G the action is clearly well dened



The Gaction U on the vector bundle H induces a natural Gaction U on the space of



continuous sections H ofH dened on H by

   

U x q U x x q

   

One should check that U x is again a section in that U x q q see

This section is evidently continuous since the Gaction on GH is continuous

There is a natural inner pro duct on the space of sections H given by

Z

   

d q q q

GH

where is the measure on GH dened by and is the inner pro duct in the b er



q H Note that dierent identications of the b er with H lead to the same inner

pro duct The Hilb ert space L H is the completion of the space H of continuous sections

c

of H with compact supp ort in the norm derived from this inner pro duct

When the measure is Ginvariantwhich is the case for example when G and H are uni



mo dular the op erator U x dened by satises

     

U x U x

HILBERT C MODULES AND INDUCED REPRESENTATIONS

When fails to b e Ginvariant it can b e shown that it is still quasiinvariant in the sense that



and x have the same null sets for all x G Consequently the RadonNiko dym derivative



q d x q d q exists as a measurable function on GH One then mo dies to

s



d x q

   

U x q U x x q

d q

Prop osition Let G be a local ly compact group with closed subgroup H and let U be a

unitary representation of H on a Hilbert space H Dene the Hilbert space L H of L sections

of the Hilbert bund le H as the completion of H in the inner product where the

c

measure on GH is dened by



The map x U x given by with denes a unitary representation of G on

L H When is Ginvariant the expression simplies to

One easily veries that the squarero ot precisely comp ensates for the lackof Ginvariance of



guaranteeing the prop erty Hence U x is isometric on H so that it is b ounded

c

  

and can b e extended to L H bycontinuity Since U xisinvertible with inverse U x

  

it is therefore a unitary op erator The prop erty U xU y U xy is easily checked



The representation U G is said to b e induced by U H



Prop osition In the context of dene a representation C GH on L H



by

  

f q f q q

 

The pair U G C GH is a covariant representation of the C dynamical system



G C GH where is given by



Given this follows from a simple computation

Note that the representation is nothing but the rightaction of C GH on



L H this rightaction is at the same time a leftaction b ecause C GH is commutative



Wenow give a more convenient unitarily equivalent realization of this covariant representation



For this purp ose we note that a section Q H of the bundle H may alternatively be

represented as a map G H whichis H equivariant in that



xh U h x



Such a map denes a section by



x x x

H



hoice where G GH is given by The section thus dened is indep endentofthec



of x x b ecause of



For to lie in H the pro jection of the supp ort of from G to GH must b e compact

c

In this realization the inner pro duct on H isgiven by

c

Z

d x x x

GH

the integrand indeed only dep ends on x through x b ecause of

Denition The Hilbert space H is the completion in the inner product of the set

ondition and the of continuous functions G H which satisfy the equivariance c

projection of whose support to GH is compact

Mackeys transitive imprimitivitytheorem

Given we dene the induced Gaction U on by

 

y U x y U x y

H

Using and as well as the denition x y xy xy xy of the

H H

Gaction on GH cf weobtain

      

U x y U x x y U xx y x y y x y

H H

Hence we infer from that



U y x y x

Replacing by in the ab ovederivation yields

s



d y x



U y x y x

d x

Similarly in the realization H the representation reads

f xf x x

H

Analogous to we then have

Prop osition In the context of dene a representation C GH on H cf



by The pair U G C GH where U is given by is a covariant



r epresentation of the C dynamical system G C GH where is given by



 

This pair is unitarily equivalent to the pair U G C GH by the unitary map V





H H given by

V x x x

H

in the sense that

 

VU y V U y

for al l y Gand

 

V f V f

for al l f C GH



Comparing with it should b e obvious from the argument leading from

to that holds An analogous but simpler calculation shows

Mackeys transitive imprimitivity theorem

In the preceding section wehave seen that the unitary representation U G induced by a unitary

representation U of a closed subgroup H G can be extended to a covariant representation

U G C GH The original imprimitivity theorem of Mackeywhich historically preceded



Theorems and states that all covariant pairs U G C GH arise in this way



Theorem Let G be a local ly compact group with closed subgroup H and consider the

C dynamical system G C GH where is given by Recal l cf that a co



variant representation of this system consists of a unitary representation U G and a representation

C GH satisfying the covariancecondition



 x

U x f U x f

x 

for al l x G and f C GH here f q f x q



Any unitary representation U H leads to a covariant representation U G C GH of



G C GH given by and Conversely any covariant representation



of G C GH is unitarily equivalent to a pairofthisform U



This leads to a bijective correspondence between the space of equivalence classes of unitary

representations of H and the space of equivalence classes of covariant representations U of

the C dynamical system G C GH which preserves direct sums and therefore irreducibility



here the equivalencerelation is unitary equivalence

HILBERT C MODULES AND INDUCED REPRESENTATIONS

The existence of the bijective corresp ondence with the stated prop erties follows bycombining

Theorems and which relate the representations of C H and C G GH with The

orems and whichallow one to pass from C H to U H and from C G GH

to U G C GH resp ectively



The explicit form of the corresp ondence remains to b e established Let us start with a technical

point concerning Rieel induction in general Using and one shows that

k V kk k where the norm on the lefthand side is in H and the norm on the righthand

obtained by Rieelinducing side is the one dened in It follows that the induced space H

from a preHilb ert C mo dule is the same as the induced space constructed from its completion

The same comment of course applies to H

We will use a gerenal technique that is often useful in problems involving Rieel induction

Lemma Suppose one has a Hilbert space H with inner product denotedby and a

linear map U EH H satisfying

U U



for al l E H

Then U quotients to an isometric map between EH N and the image of U in H When

the image is dense this map extends to a unitary isomorphism U H H Otherwise U is

unitary between H and the closure of the image of U

In any case the representation C E B is equivalent to the representation C E B

denedbycontinuous extension of

AU U A I

It is obvious that N kerU so that comparing with one indeed has U

U

We use this lemma in the following way To avoid notational confusion we continue to de

note the Hilb ert space H dened in Construction starting from the preHilb ert C mo dule

C G
C H dened in the pro of of by H The Hilb ert space H dened b elow

c c

however will play the role H in and will therefore b e denoted by this symbol

Consider the map U C G H H dened by linear extension of

c

Z

U v x dh xhU hv

H

Note that the equivariance condition is indeed satised by the lefthand side as follows

from the invariance of the Haar measure

Using and with G H one obtains

Z Z Z

dx dh dh U h v U hw xxhv U hw v w

R

L G



G H H

cf On the other hand from and one has

Z Z Z

xk xhU k v U hw dh d x dk U v U w

H

H GH H

Shifting h kh using the invariance of the Haar measure on H and using one veries

It is clear that U C G H is dense in H soby Prop osition one obtains the

c

desired unitary map U H H

Using and one nds that the induced representation of C G GH onH is

given by

Z



dy f xy x y f x

H G

this lo oks just like with the dierence that in lies in C G whereas in

c

lies in H Indeed one should check that the function f dened by satises

the equivariance condition

Finally it is a simple exercise the verify that the representation C G GH dened by

corresp onds to the covariant representation U G C GH by the corresp ondence



of Theorem

Applications to quantum mechanics

quantum mechanics The mathematical structure of classical and

In classical mechanics one starts from a phase space S whose p oints are interpreted as the pure

states of the system More generally mixed states are identied with probability measures on S

The observables of the theory are functions on S one could consider smo oth continuous b ounded

measurable or some other other class of realvaued functions Hence the space A of observables

R

may b e taken to b e C S R C S R C S R or L S R etc

 b

There is a pairing h i SA R between the state space S of probability measures

R

on S and the space A of observables f This pairing is given by

R

Z

h f i f d f

S

The physical interpretation of this pairing is that in a state the observable f has exp ectation

value h f i In general this exp ectation value will b e unsharp in that h f i h f i However

in a pure state seen as the Dirac measure on S the observable f has sharp exp ectation value

f f

In elementary quantum mechanics the state space consists of all density matrices on some

Hilb ert space H the pure states are identied with unit vectors The observables are taken to

b e either all unb ounded selfadjoint op erators A on H or all b ounded selfadjoint op erators or all

compact selfadjoint op erators etc This time the pairing b etween states and observables is given

by

h Ai Tr A

In a pure state one has

hAi A

Akey dierence b etween classical and quantum mechanics is that even in pure states exp ectation

values are generally unsharp The only exception is when an observable A has discrete sp ectrum

and is an eigenvector of A

In these examples the state space has a convex structure whereas the set of observables is a

real vector space barring problems with the addition of unb ounded op erators on a Hilb ert space

Wemay therefore saythataphysical theory consists of

a convex set S interpreted as the state space

arealvector space A consisting of the observables

R

a pairing h i SA R which assigns the exp ectation value h f i to a state and

R

an observable f

In addition one should sp ecify the dynamics of the theory but this is not our concern here

The situation is quite neat if S and A stand in some duality relation For example in the

R

classical case if S is a lo cally compact Hausdor space and we take A C S R then the

R 

space of all probability measures on S is precisely the state space of A C S in the sense of



Denition see Theorem In the same sense in quantum mechanics the space of all

density matrices on H is the state space of the C algebra B H of all compact op erators on H 

APPLICATIONS TO QUANTUM MECHANICS

see Corollary On the other hand with the same choice of the state space if we take A

R

to b e the space BH of all b ounded selfadjoint op erators on H then the space of observables

R

is the dual of the linear space spanned by the state space rather then vice versa see Theorem

In the C algebraic approach to quantum mechanics a general quantum system is sp ecied

by some C algebra A whose selfadjoint elements in A corresp ond to the observables of the

R

theory The state space of A is then given by Denition This general setting allows for the

R

existence of sup erselection rules We will not go into this generalization of elementary quantum

mechanics here and concentrate on the choice A BH

Quantization

The physical interpretation of quantum mechanics is a delicate matter Ideally one needs to sp ecify

the physical meaning of any observable A A In practice a given quantum system arises from a

R

classical system by quantization This means that one has a classical phase space S and a linear

 

map Q A LH where A stands for C S R or C S R etc and LH denotes some



R R

space of selfadjoint op erators on H suchasB H or BH Given the physical meaning of

 R R

a classical observable f one then ascrib es the same physical interpretation to the corresp onding

quantum observable Qf This provides the physical meaning of al least all op erators in the

image of Q It is desirable though not strictly necessary that Q preserves p ositivityaswell as

the approximate unit



It is quite convenient to assume that A C S R which choice discards what happ ens at



R

innityonS We are thus led to the following

Denition Let X bealocal ly compact Hausdor space A quantization of X consists of

a Hilbert space H and a positive map Q C X BH When X is compact it is required that



I and when X is noncompact one demands that Q can be extended to the unitization Q

X

C X by a unitpreserving positive map

 I

Here C X and BH are of course regarded as C algebras with the intrinsic notion of



p ositivitygiven by Also recall Denition of a p ositive map It follows from that

a p ositive map automatically preserves selfadjointness in that

Q f Qf

for all f C X this implies that f C X R is mapp ed into a selfadjoint op erator

 

There is an interesting reformulation of the notion of a quantization in the ab ove sense

Denition Let X be a set with a algebra of subsets of X A p ositiveop erator



valued measure or POVM on X in a Hilbert space H is a map A from to BH

P

A for the set of positive operators on H satisfying A AX Iand A

i i i

i

any countable col lection of disjoint where the innite sum is taken in the weak operator

i

topology

A pro jectionvalued measure or PVM is a POVM which in addition satises A



A A for al l

 

Note that the ab ove conditions force A I A PVM is usually written as E

it follows that each E is a pro jection take in the denition This notion is familiar



from the sp ectral theorem

Prop osition Let X bealocal ly c ompact Hausdor space with Borel structure Thereis

a bijective correspondencebetween quantizations Q C X BH andPOVMs A on



S in H given by

Z

Qf dAx f x

S

The map Q isarepresentation of C X i A is a PVM 

Stinesprings theorem and coherent states

The precise meaning of will emerge shortly Given the assumptions in view of and

wemayaswell assume that X is compact

Given Q for arbitrary H one constructs a functional on C X by f

Qf Since Q is linear and p ositive this functional has the same prop erties Hence the

Riesz representation theorem yields a probability measure on X For one then puts

A dening an op erator A by p olarization The ensuing map A

ed to have the prop erties required of a POVM is easily check

Conversely for eachpair H aPOVM A in H denes a signed measure

on X by means of A This yields a p ositive map Q C X BH by

R

Qf d x f x the meaning of is expressed by this equation

X

Approximating f g C X by step functions one veries that the prop erty E E is

equivalenttoQfg Qf Qg

Corollary Let A beaPOVM on a local ly compact Hausdor space X in a Hilbert

space H There exist a Hilbert space H a projection p on H a unitary map U H pH



and a PVM E on H such that UAU pE p for al l

Combine Theorem with Prop osition

When X is the phase space S of a physical system the physical interpretation of the map

A is contained in the statementthatthenumber

p Tr A

is the probability that in a state the system in question is lo calized in S

When X is a conguration space Q it is usually sucienttotakethe p ositivemapQ to b e

a representation of C QonH By Prop osition the situation is therefore describ ed bya



PVM E on Q in H The probability that in a state the system is lo calized in Q

is

p Tr E

Stinesprings theorem and coherent states

By Prop osition a quantization Q C X B H is a completely p ositive map and



Denition implies that the conditions for Stinesprings Theorem are satised We

will now construct a class of examples of quantization in which one can construct an illuminating

explicit realization of the Hilb ert space H and the partial isometry W

Let S b e a lo cally compact Hausdor space interpreted as a classical phase space and consider

an embedding of S into some Hilb ert space H such that each has unit norm so that

a pure classical state is mapp ed intoapurequantum state Moreover there should b e a measure

on S such that

Z

d

 

S

for S for all H The are called coherent states



Condition guarantees that wemay dene a POVM on S in H by

Z

A d



where is the pro jection on to the onedimensional subspace spanned by in Diracs notation

one would have j j

The p ositive map Q corresp onding to the POVM A by Prop osition is given by

Z

Qf d f

S

In particular one has Q I S

APPLICATIONS TO QUANTUM MECHANICS



For example when S T R R sothat p q one maytake

ipq ipx

pq n xq

x e e

in H L R Eq then holds with dp q d pd q Extending the map Q from

C S toC S in a heuristic way one nds that Qq andQp are just the usual p osition and

 i i

momentum op erators in the Schrodinger representation

In Theorem wenowputA C S B BH AA for all A We may then



erify the statement of the theorem by taking v

H L S d

The map W HH is then given by

W

It follows from that W is a partial isometry The representation C S is given by



f f

Finally for one has the simple expression



Qf U Qf U pf p

Eqs and form the core of the realization of quantum mechanics on phase space

One realizes the state space as a closed subspace of L S dened with resp ect to a suitable

measure and denes the quantization of a classical observable f C S asmultiplication by f



sandwiched b etween the pro jection onto the subspace in question This should b e contrasted with

the usual way of doing quantum mechanics on L Q where Q is the conguration space of the

system

In sp ecic cases the pro jection p WW can b e explicitly given as well For example in the

p

case S T R considered ab oveonemay pass to complex variables by putting z q ip

We then map L T R d pd q into K L C d zd z expz z i by the unitary

op erator V given by

p p

z z

z e z z q z z V z p



One maythenverify from and that VpV is the pro jection onto the space of entire

functions in K

Covariant lo calization in conguration space

In elementary quantum mechanics a particle moving on R with spin j N is describ ed by the

Hilb ert space

j

H L R H

j

QM

j 

where H C carries the irreducible representation U SO usually called D The basic

j j j

S S

physical observables are represented byunb ounded op erators Q p osition P momentum and

k k

S

J angular momentum where k These op erators satisfy the commutation relations

k

say on the domain S R H

j

S S

Q Q

k l

S S

P Q i

kl

k l

S S S

J Q i Q

klm

k l m

S S

P P

k l

S S S

J J i J

klm

k l m

S S S

P J P i

klm

m k l

Covariant lo calization in conguration space

justifying their physical interpretation

The momentum and angular momentum op erators are most conveniently dened in terms of a

j j

unitary representation U of the Euclidean group E SO nR on H given by

QM QM

j 

U R aq U R R q a

j

QM

S

In terms of the standard generators P and T of R and SO resp ectively one then has P

k k

k

j j

S

idU P and J idU T see The commutation relations follow

k k

QM QM

k

utation relations in the Lie algebra of E from and the comm

j j

Moreover we dene a representation of C R onH by



QM QM

j

f f I

j

QM

where f is seen as a multiplication op erator on L R The asso ciated PVM E on R

j

in H see is E I in terms of which the p osition op erators are given by

QM  j

R

S

Q dE xx cf the sp ectral theorem for unb ounded op erators Eq then reects the

k

k

R

j

utativityofC R as well as the fact that is a representation comm



QM

Identifying Q R with GH E S O in the obvious wayonechecks that the canonical

leftaction of E on E S O is identied with its dening action on R It is then not hard to

j j

verify from that the pair U E C R is a covariant representation of the C



QM QM

dynamical system E C R with given by The commutation relations



are a consequence of the covariance relation

S S S

P and J and their commutation rela Rather than using the unb ounded op erators Q

k k k

j j

tions we therefore state the situation in terms of the pair U E C R Such a



QM QM

j

pair or equivalently a nondegenerate representation of the transformation group C algebra

QM

C E R cf then by denition describ es a quantum system which is lo calizable in R

j

and covariant under the dening action of E It is natural to require that b e irreducible

QM

in which case the quantum system itself is said to b e irreducible

and covariant under Prop osition An irreducible quantum system which is localizable in R

E is completely characterized by its spin j N The corresponding covariant representation

j j

U E C R given by and is equivalent to the one described



by and

j

This follows from Theorem The representation U E dened in is unitarily

QM

j j

equivalent to the induced representation U To see this checkthat the unitary map V H

j j

j j j

H dened by V q e q intertwines U and U In addition it intertwines the

QM QM

j

representation with as dened in

QM

This is a neat explanation of spin in quantum mechanics

Generalizing this approach to an arbitrary homogeneous conguration space Q GH a non

degenerate representation of C G GH on a Hilb ert space H describ es a quantum system

which is lo calizable in GH and covariant under the canonical action of G on GH By this

is equivalent to a covariant representation U G C GH on H and by Prop osition



one may instead assume one has a PVM E on GH in H and a unitary representation

U G which satisfy



U xE U x E x

for all x G and cf The physical interpretation of the PVM is given by

the op erators dened in play the role of quantized momentum observables Generalizing

Prop osition wehave

Theorem An irreducible quantum system which is localizable in Q GH and covariant

The under the canonical action of G is characterizedbyanirreducible unitary representation of H

j

system of imprimitivity U G C GH is equivalent to the one described by and



APPLICATIONS TO QUANTUM MECHANICS

This is immediate from Theorem

For example writing the twosphere S as SOS O one infers that SOcovariant quan

tum particles on S are characterized byaninteger n Z For each unitary irreducible represen

tation U of SO is lab eled by suchannandgiven by U expin

n

Covariant quantization on phase space

Let us return to quantization theory and ask what happ ens in the presence of a symmetry group

The following notion which generalizes Denition is natural in this context

Denition A generalized covariant representation of a C dynamical system

G C X where arises from a continuous Gaction on X by means of consists of a



pair U Q where U is a unitary representation of G on a Hilbert space H andQ C X BH



is a quantization of C X in the sense of Denition which for al l x G and f C X

 

satises the covariancecondition

U xQf U x Q f

x

This condition may b e equivalently stated in terms of the POVM A asso ciated to Q

cf by



U x Ax U xA

Every ordinary covariantrepresentation is evidently a generalized one as well since a rep

resentation is a particular example of a quantization A class of examples of truly generalized

covariant representations arises as follows Let U G C GH b e a covariant representation



on a Hilb ert space K and supp ose that U G is reducible Pick a pro jection p in the commu

tantof U G then pU Gpp is a generalized covariant representation on H pK Of course

U is describ ed by Theorem and must b e of the form U This class actually turns

out to exhaust all p ossibilities What follows generalizes Theorem to the case where the

representation is replaced byaquantization Q

Theorem Let U G QC GH be a generalized covariant representation of the C



ct to the canonical Gaction on GH dynamical system G C GH dened with respe



There exists a unitary representation U H withcorresponding covariant representation U

of G C GH on the Hilbert space H as described by and and a pro



jection p on H in the commutant of U G such that pU Gp p and U G QC GH



areequivalent

We apply Theorem Toavoid confusion we denote the Hilb ert space H and the repre

sentation in Construction by H and resp ectively the space dened in and the

induced representation will still b e called H and as in the formulation of the theorem

ab ove Indeed our goal is to showthat H maybeidentied with H Weidentify B

in and with BH where H is sp ecied in we therefore omit the representation

o ccurring in etc putting H H

For x G we dene a linear map U xonC GH H by linear extension of



x U xf f U

x

Since and U is a representation U is clearly a Gaction Using the covariance

x y xy

condition and the unitarityof U x one veries that

U xf U xg f g

 

where is dened in Hence U G quotients to a representation U GonH Com



puting on C GH H and then passing to the quotient one checks that U isacovariant



representation on H By Theorem this system must b e of the form U up to unitary

equivalence

Finally the pro jection p dened in commutes with all U x This is veried from

and The claim follows

LITERATURE

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